Is it bad form to let the constant of integration equal several different values over a calculation

It is considered bad form. The usual approach, at least in a final presentation (as opposed to when you're actually doing the calculations), is to know how many different $C$s you need, and use either $C',C''$, etc. until you get to the final one, which is just $C$ (if there aren't too many, $C'''''$ is a bit ridiculous), or use indices: $C_1,C_2,$ etc.

If what you're writing is only meant for your eyes, you can really do whatever you feel like; mathematical notation conventions are there to facilitate communication between people, not to put restrictions on what people write down as their own personal notes. That being said, eliminating a potential source of confusion at nearly no cost of writing speed or cognitive load sounds like a good thing to me, so I would suggest you use something like this in those cases as well.

As Arthur's answer has pointed out, in general you should clarify that your variable is changing. However, in this particular situation (solutions to differential equations) it is very commonly understood that C isn't meant to be understood as a normal variable, but rather as a placeholder to mean "some constant" (often called a generic constant). This is of course a fundamental departure from normal equations, but is so useful in situations where you would need way too many different C's that it is often accepted and understood. As ever with notation, consider who your audience will be and what they are used to, and clarify when necessary.

Arthur's answer is excellent, but I think you'd find it useful to see a situation with a previous question on math.stackexchange where re-using the same variable to represent constants lead to much more subtle problem. The question is here.

Even though they (correctly) used $C_1$ and $C_2$ to represent arbitrary constants when solving a diff-eq, they re-used those same variables for another function that was related to the first function through some equations, and ended up with an incorrect final answer. (My answer giving a full description of what went wrong is here. )