Only a much better, non-lame ability, which is much better than those lame-little-bards do.
Obligatory
bards manipulate the echoes of creation examples.
Bards start at 1d6 and work up to 1d12 while your chart starts at 1d4 and works up to 1d12, and increases at ealier levels for the bard. By 5th level bards are using about a dozen dice per day while your chart is up to 3 per character for the day. Other than that the bonuses are to the same rolls as the bardic inspiration.
Well, he might inspire one of his companions, any way, for one roll.
He might do that at 1st level with a 13 CHA or lower. This only happens if the character isn't using the ability, and if they don't use the bard ability I'm not sure why they would choose to use hit dice implementation of the same ability over healing based on what you are telling me they are doing now.
First, the differences between proficiency at lower levels and the highest levels is not sufficient. If you compare two PCs with proficiency +2 and +6 in a contested check, the lower one has to great a chance to win. Now, in our current game we've expanded the top bonus to +8, which makes it a bit better without having to adjust too many other things.
I was looking for a different way to handle it. Spending a HD to get a bonus would allow two things:
1. the higher level PC would get a larger potential bonus over the lower level one
2. the higher level PC can tap into the resource more often than the lower level one
Originally, I was just going to use the HD as the bonus itself, as you suggest, but then the die-type could create the bigger bonus, instead of it level-based, which is why I chose proficiency instead.
The issue with giving more inspiration is it doesn't help in those cases where getting a higher number is more important than the chance of getting a better number, if that makes sense?
Finally, I would like to tie it into an existing system instead of adding a new one, such as Hero points. By using HD, I am forcing the player to consider the reward of a higher result now vs the risk of not having the HD later to restore hp.
You are trying to contradict the game premise that lower level / CR is more relevant longer as PC's level that 5e embraces. The bard part is actually irrelevant (because you are trying to rescale the proficiency range) but I find it funny you stated the similar bonuses to the same rolls are lame in one example and not lame in another.
What will happen is NPC's will not need to use their hit dice for healing or anything else so it's an additional resource your players either won't use because of the competition for healing and because they seem to have demonstrated they don't use a similar ability already, or they will use it and run out of a resource the opponents will still have available. In both cases it just ends up being a buff for your antagonists.
The alternative is not to give the ability to the antagonists, in which case it ends up being nothing but a buff to players.
What's really causing your opposed check issue is you want the d20 to have less impact on the results of the opposed checks. An easier way of doing that would be to change your resolution method to a multiple-check approach. That brings it statistically closer to average the more dice are rolled. DM's already do that with a more complex series of checks sometimes so you might want to do that more.
Your explanation doesn't make it clear on why the bonus is applying to attack rolls and saving throws since both are static instead of opposed. It looks like you want level to matter more. Barring automatic hit/miss, this illustrates what you are doing now:
I included +3, +4, and +5 ability score modifiers at appropriate levels. It's not just proficiency bonus that matters here. It's proficiency, ability, and abstract circumstance / luck. The ability score significance moderates the die roll and what you are doing.
Here's that illustration for DC's with the full range of bonuses looks like:
Both charts include +7/+8 proficiency and cap out at +5 ability score. Neither includes expertise but the general idea is shown. With the +6 proficiency cap the check using a primary attribute the character is 30% high vs low and 60% high vs low without assuming a primary stat.
I'm bringing this up because as you increase the value of your bonuses you decrease the value of your ability scores in addition to the value of the dice. It's something you and your players might want to discuss if you want to devalue the relative contribution of the ability score as much as would be necessary to make level more relevant.
The comparison also demonstrates the value of targeting weak attributes. The gap from a lower level to a higher level is because there isn't much change in going from 16 STR to 20 STR. If the proficiency scale goes to 10 then it counts twice as much as ability score, for example. That comes with growing pains as you probably noticed with your current spread.
Circling back to the opposed check, a low level character with proficiency and 16 ability score in an opposed check against your +8 proficiency and a 20 ability score (both focuses on the same skill) looks like this:
The high proficiency character should be winning opposed checks the vast majority of the time so it seems like there might be some anecdotal bias clouding judgement here unless I'm missing something.
For comparison, this is a 10 ability score +8 proficiency versus 16 ability score +2 proficiency.
Your spread should have proficiency beating low level proficiency and ability score 2 out of 3 times.
Here is what both are the comparisons for standard +6 proficiency so you can see the impact your extra proficiency makes.
When comparing a high attribute to a high attribute the high proficiency wins the opposed check 7% more often, so it matters about 1 in 14 checks.
High proficiency with no bonus drops 9% because proficiency his a bit more challenge overcoming ability. This is the low end of high proficiency versus low level.
In the first two charts, ability score made an 18% range of wins based on 0 ability score bonus, and in the second set that increased to a 20% range of wins based on ability score bonus for the high proficiency character.
Note that the only room the low proficiency character has it so go down in ability score and might not actually have that proficiency bonus. For example, if this was broken stealth vs perception not all low level characters have 16 WIS even if many of them do take perception. That increases the success rate for high level proficiency over all.
I haven't gone on a tangent here. I'm demonstrating how much difference proficiency makes going from low to high level. What the proposed bonus does is gives that option to make a save or not fizzle an attack when it matters. The frequency you are giving them out won't change your over-all spread much but you will have the same benefits of having a bard in the party when you don't have a bard in the party. ;-)
I'm more curious about who is the high level and who is the low level. As I mentioned above, the opponents will have a fresh resource in which the party is limited. If the party is higher level it's a bonus to the lower level opponents when the party is out or not willing to use this resource. If the party is facing higher level opponents the same risk occurs where it's just a big bonus for the big bad after the party ran out earlier, or the party didn't use the option at all saving up for the big bad who has bigger bonuses as theoretically higher level.
That's quite a bit of feedback and it's late here. Hope it helps.