The call belongs to the base umpire, BU. An "FU" is how we refer to a missed call. (For those with sensitive demeanors, that's a "foul-up".)
If the BU wants to discuss it with their partner, either because a coach has asked a question, or because the BU just wants to, the BU can. If after that discussion, the BU wants to change the call, it is allowed.
The coach has no right to require the BU to seek assistance. A coach's dissatisfaction in a judgment call does not constitute the right to protest. Protests are for misapplications of the specifics of a rule.
In your scenario, my concern would be the PU discussing with the coach a call by the BU. If I had been the BU, I would have stated to the to coach, "That was my call, if you've got a question about it, talk to me." After the game, I would have repremanded my partner.
"Appeals" are for base running infractions that must be ruled on, e.g. a runner missed touching a base or left early on a caught fly ball. They are not for overriding judgment. Some organizations instruct their umpires to always ask for help on a checked swing. I think this is the only time that 'requesting' help is dictated. I do not believe ASA requires this on a checked swing, but most good umpires always do.
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Dan
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