You should be able to - at any time. I panicked and hired the wrong coach for my bullpen during the first day of negotiations and now I will be stuck with a lacklustre coach all season when I have the money left over to pay for a better one. It's unfair and unrealistic. If you're struggling half way through a season a coaching change can be a difference maker. Just my opinion.

Coaches affect development not game play.    They will not help you if you're struggling.

Bench coaches affect in-game play when the manager is ejected.  As per the "Help" page.

In theory.   Not sure how many times a BL manager is ejected and, when he is, what sort of "crazy" moves a bad one will make in your ejected absence.   Pull the starter early?   Use the wrong set-up man?   Put a catcher at SS?    I'd need to see some concrete evidence that the bench coach goes out of his way to hijack your settings once he's running the show. 

I'd be able to produce some questionable moves that seem to go against common sense without an ejection so you'd really need to see an RP go to 2B with a back-up 2B on the bench to prove the bench coach does something out of control. 

I can respect and understand their use in regards to player development over gameplay but I feel it is less realistic to it have the ability to make changes to the coaching staff at any time. Once you pay a coach for a season they are entitled to all of their salary but if a mistake is made or change wanted then you should be able to do it.

My problem is I rushed my coach signings and lost my bullpen coach somehow, so hired a AAA coach of mine to do the job and now that will definitely affect my player development. I made a mistake but if I have the funds available I should be able to fix it. Most of my bullpen is 25 or younger and my bullpen coach only has 50 patience.

I'll explain why it's in place and how it COULD be resolved(but probably won't).

There are always jerks on the internet.   Some of them get their jollies by attempting to destroy any fun someone else might be having.   So, JoeJerk gets mad at his leaguemates because they vetoed his trade.   He fires all his minor league coaches and hires the worst possible replacements.   The entire minor league system is trashed as players decline rather than development.   It creates a situation that can't be rectified for several seasons and that team is tough to fill.

Now, to negate that possibility, WifS could make it where you have to hire a BETTER coach before firing the current one.   However, that would require new coding for a select few situations like yours.    That said, at the BL level, it's unlikely you have anyone who's going to develop much anyway.   Looking at your roster, Gandarillas is the only one I'd expect to move much.   And he's a 25 y/o 3rd year pro.   He's mostly done developing.