This looks to be entirely up to DM discretion.
There are two equally valid positions on this and the rules do not refute either one.
The first is that this is a completely separate spell, and slot, and that the two have no overlap with your casting class. The spell you learned and the slot you obtain are kept separate from the class spells if you have levels in that class.
The second position is that it is a class spell, from your class list and you can retrain it as you would when you take a new level in that class.
The second position is mildly problematic in that you can continue to obtain higher level spells with the retraining, and it may lead to this MC being simply the obtaining of another 1st level spell slot, rather than feeling like a mini MC.
The first position is problematic in that it introducing additional bookkeeping and may be perceived to be overly harsh and pedantic.
In my games, I think I would allow the MC feat to simply be an additional first level spell slot and allow the spell to be retrained if it's from a class that allows that and you take levels in that class. I would probably not make them remember which spell goes with their MC beyond that the extra slot must be a spell from that class. This may be too lenient for some games though.
Sometimes, rewards and incentive are motivational enough on their own. In a tabletop RPG like this where bonus exp, gold, or any other non mundane item gifted to a player for doing something that is largely required by the player as a means of character maintenance after each session and before the next is going to lead to unfair advantages when 1 or 2 of your average group of 5 end up majorly ahead in exp or currency or wealth measured in possessions (items obtained). Even Inspiration is something not to be handed out lightly.
So when rewards and incentives create imbalance? What's left is rules and regulations. My group always does their leveling at the end of each session, after XP is awarded, if said XP is enough to progress them to the next level. Sometimes our sessions run for 5 or 6 hours and everyone is too tired to do all the math and additions required for level up. No worries, they have a full 7 days to level their character! Seven days is plenty of time. There should be time in anyone's schedule over the course of 7 days to level up a character. Minds can be slippery sometimes, however, and reminders are required.
So how do you effectively go about this while respecting your players time restrictions and motivate them in the process? You make a hard rule. Session start time is Session Start Time . Character leveling up is the players responsibility. Anyone not sufficiently leveled up by the start of the next session 7 days later uses the character they have as represented on their character sheet. If they didn't go buy manacles and a chain, and the start of your session doesn't allow for shopping? If they didn't add all the nifty new things they gained for reaching level 4 and the session is starting? Well they're missing out.
You might say "Well that seems harsh, Airatome, where does the madness end!?" Well; you supplement that with reminders, if possible. Our group plays over skype so there is a consistent group chat always active. Whatever forum you use to bring your players together as a group has a method of messaging your group. So remind them at the end of each session to level up and shop. Remind them a few days later to level up and shop. Remind them the day before your next session to level up and shop. Anyone failing their responsibilities to keep their character sheet current after positive reminders and reinforcement is then stuck adhering to the rule of Session Start Time.
They can then level up their character if there's a break, or after the end of the session, and catch up to where they belong. Additionally, time allowing, you COULD have a soft session start time 1 hour before the implemented hard start time for those who need to catch up in levels and spend gold and such. This will not only give the people who already have done so time to make it to the session's actual start time, allow you as the DM to help any players who need it with their level up and HP rolls, but also allows everyone to start on an even note after the soft hour has concluded and the actual session begins. No delays. No giving out gold or items or Inspiration to those who should earn them in the game not out of the game because of a character sheet.
Regardless of the optional 'soft start time' being used, implementing a 'Hard Start Time' along with positive reminders after, in between, and a day before sessions should motivate them in a way that does not sow discord or advantages/disadvantages among the players; because after all is said and done, they DO gain the level they were missing; it's simply their responsibility, not yours, to keep their character current.
Best Answer
There are not. The game assumes that all characters are taking care of their equipment (and studying and practicing and so on) off-screen, and basically assumes that there are never any particular troubles with this. The rules work this way because the authors assume that the characters have much bigger, more interesting troubles to be concerned about.
The only way we’re likely to see such rules from WotC is, maybe, as part of some limited optional variant rule. Such variant rules are rarely comprehensive or complete; rather they just offer a few ideas for how you might change the game some, and then expects DMs to figure out the details.
Which is what your DM here is doing. He is attempting to incorporate a (realistic) detail that the rules themselves ignore. The rules themselves encourage that kind of customization, but it is then the responsibility of the DM to fully flesh out those rules and clearly explain them to the players (at least insofar as the player characters would know them).
I would further add that it is his responsibility to fully understand the situation: why the rules are the way they are, how this change is going to affect things, how other things should change for consistency, and what the effects those will have, and so on. Homing in on one particular detail while ignoring the rest is a mistake that leads to an inconsistent game. And changing everything to pay greater attention to maintenance is a huge change with significant effects, and by definition is also taking away attention from elsewhere. The DM should be aware of all of this when making changes.
So you will have to ask the DM for the rules he’s making up. I would suggest, at the very least, that a magic bow should be exempt from such concerns; magic items are generally far, far tougher than their non-magic counterparts, and such an exemption would mean that dedicated archers should be able to cease to worry about this problem at relatively low levels.
Why isn’t this part of the game?
What follows is a somewhat-lengthy, somewhat-tangential discussion of why such a rule is not part of the game, and why I say it almost-certainly never will be (outside of optional variant rules). This may be of interest to you, and it may also be something you would be interested in showing to your DM (such is my hope, at any rate). As I’ve already stated, the DM should be taking on a lot of responsibility for understanding the game and understanding the significance of his changes when he makes them. This may help him understand that.
Despite some claims, D&D is not a be-all, end-all system perfect for every sort of game, or even every sort of fantasy game, or even every sort of medieval fantasy game. Its rules, in aggregate, paint the picture of a very certain sort of fantasy game, which come from its roots and is hinted at by the name. In essence, D&D is first-and-foremost designed to be for dungeon-delving and dragon-slaying.
Now, to be clear, this priority may stem from D&D’s history, but that does not mean the game hasn’t changed over the years or across the editions. What it means to go out dungeon-delving or dragon-slaying has changed some—and Wizards of the Coast has tended more-or-less to try to stick to modern expectations for those activities rather than necessarily trying to hew as close as possible to the original vision. 5e is something of a reboot for them, after 3e (perhaps unintentionally) strayed from those ideals and 4e embraced that straying to a degree that alienated significant portions of the audience, but nonetheless it has plenty of modern sensibilities to it.
The reason I bring this up is because this kind of maintenance concern is precisely the kind of thing that (modern, WotC) D&D has absolutely and categorically ignored. It is not a part of the fifth edition, and that choice seemed very much intentional on Wizards’ part. It’s impossible to say if that reflects some market data they have or simply the personal preferences of their team, but the fact remains that 5e has set its abstraction threshold firmly above the point where one would be concerned about an always-strung bow.
“Abstraction threshold” is an important term in game design, and refers to the degree to which the game attempts to emulate reality. All games, by definition, must simplify reality to be playable, and simplification is often done through abstraction—conflating things that are really distinct, ignoring caveats, qualifications, or limitations, smoothing out edge cases. This is important to making a game; where you choose your abstraction threshold defines a whole lot about what kind of game you have and what sort of stories you can tell playing it.
And, to quote Heinlein, there ain’t no such thing as a free lunch: changing the abstraction threshold focuses more attention on one thing, at the cost of other things. Attention and abstraction is not strictly zero-sum, but everyone’s time and attention is limited. Every rule demands time and attention, so the more demands you have in one area, the less will be possible to devote to another.
WotC D&D’s abstraction threshold is set to encourage “epic” narratives. Caring for bowstrings is not something that is part of that narrative. Legendary bow shots are what that narrative wants to relate. Having rules about bowstring maintenance draws attention away from epic bowshots—time and attention must be paid to maintaining the bowstring, and that will necessarily come from time that could have been spent setting up, attempting, and making legendary bow shots. And so I assert that bowstring care will never find its way into the rules of the game. Other games have different abstraction thresholds, and even earlier editions of D&D may have had much greater focus on this kind of maintenance and preparation, and may very well have rules for damage to a bowstring if left strung.