A giant spider is breaking the normal rule that your Ranger's Companion must be CR 1/4 or less, so I'm not sure if your GM is still using the clause from the Ranger's Companion feature...
"Add your proficiency bonus to the beast’s AC, attack rolls, and damage rolls, as well as to any saving throws and skills it is proficient in."
...Normally a giant spider has AC 14 (3 points from Dex, 1 point from natural armor). But if this rule is still in effect (check with GM) then your proficiency bonus would be added. At 1st to 4th level, your proficiency bonus is +2. Which would give it AC 16.
I think OP might be using the Primal Companion rules from Tasha's, since as you point out, the rules details given here don't make sense otherwise. At 3rd level, a "giant spider" with the Beast of the Land statblock would have AC 15.
If that's the case, then OP would probably want to use medium barding, not light barding, since the Beast of the Land only has +2 DEX. Scale mail barding would get the spider's AC up to 16; however, it'd cost 200 GP, and Muffin would get equivalent AC just through normal progression at 5th level anyway, so it may not be worth it financially. Half-plate barding's not a good option for similar reasons, IMO, since it'd cost 3,000 GP and would be obsolete at 9th level, all for a fairly minor benefit.
So, barding's kind of a mixed bag. Taking Mounted Combatant only works if OP's character is Small, since the Beast of the Land is medium. Barksin's obsolete by the time OP would get it, similar to barding, and Warding Bond and Shield of Faith aren't ranger spells, and also expensive in terms of HP/concentration/spell slots.
Overall, if OP's using the Tasha's version, there's not really a lot that can be done to improve the companion's survivability. However, that's kind of baked into the design, since the companion can be revived easily by spending a spell slot. The only concern there is the hour limit for raising the companion. However, getting Gentle Repose via the Ritual Caster feat would potentially solve that issue. (The wording's unclear, but I suspect most DM's would allow it.)