That's alot of if's and none of which are made explicitly clear in the OP. The other thing I haven't seen discussed is the fact that it very much steps all over the themes, niche and schtick of the divine based classes... How does the cleric, or traditional paladin feel about the fact that once the DM makes this decision the fiction is now established that anyone can heal or gain the direct attention of gods for miracles just by saying a prayer?
I don't actually have a Cleric or Paladin proper in my home DW game, though the Druid and Ranger both took the option that lets them pick up Cleric spellcasting (though the Druid semi-retired upon actually going to that point.)
But let's say I did have one. There are only four possible situations I can think of where this could apply:
1. The Cleric or Paladin
is the dying friend. The Fighter is showing an impressive display of piety, and more importantly, showing just how important the dying character is to them. Far from stepping on the divine character's toes, this sounds like a beautiful opportunity for these two characters to become much closer. The Cleric or Paladin could become a mentor in the faith to the Fighter. Or, if my aforementioned "you rolled badly and now you're stuck in a deal with a dark power" thing applies, well now we have some juicy drama, don't we? Will the Fighter come clean and try to get the Cleric/Paladin to help them? Will they keep it secret? Will there be marks on the Cleric/Paladin (visible or invisible) which could negatively affect their future situation? Super uncertain.
2. The C/P is incapable of helping for whatever reason. Similar to the above, but less poignant and personal. Same overall pattern applies though.
3. The C/P could help, but is refusing to. This creates tension rather than closeness, but is essentially the antagonistic version of the previous two options.
4. The C/P can help, is willing to help, and would do so if allowed the opportunity, but the Fighter jumped in front and is doing this first.
Of these, only the last is one where I see any issues...and it's unclear why the Fighter would do this at all. Surely, if there's an issue like this and nothing preventing it, the Cleric or Paladin would have just used magic (spells, Lay on Hands) to fix the problem. Or, if their magic has run dry, it still seems weird that the Fighter would attempt an untrained thing when the Cleric or Paladin is right there and fully capable of doing the same thing with higher effectiveness.
I guess what I'm saying is, we necessarily need to assume actively crappy behavior on the part of the Fighter in order to get to the notion that the Fighter is stepping on thematic toes. Otherwise, we have a situation where the thematically-best-fit character is unable or unwilling to help, and thus someone else is trying to fill that intentional or situational void.
More importantly, any concerns a player might have about "well why can't I do that?" can be answered very simply: "You already can. You have no risk and a large supply of readily-available power. A standing deal, if you will. You could always try to negotiate for extra stuff, if you like, but that will have risks like, but not the same as, what the Fighter risked. You'll probably be better at it, and the kinds of results you get will generally be more favorable. But asking for extra beyond what you are already promised is something that should have consequences."