I've thought about this aspect, but the thing is, how would you ever even show you were refused without a recording? (Say if you're in a 2-party-consent state? Which they would obviously not consent to?)

Civil cases are based on preponderance of evidence, and your testimony is evidence. If they refute it and they're lying, that's a felony, and I don't think these professionals are being paid enough to commit felonies for their employer.

This reminds me of another question I've had -- does anyone ever get prosecuted for false testimony? I feel like I don't really hear of this happening, which has always baffled me.