'Let me check and come back to you' turns a reflex yes into a decision — then one short line, then the same line again. Plus what to do with the guilt that turns up regardless.
Which is why better phrases don't help — you never reach the moment where you'd use one. 'Let me check and come back to you' isn't a refusal, it's the pause where a decision becomes possible.
Under pressure you don't browse options, you reach for whatever's automatic. One short line you'd genuinely say, learned properly, beats a library you'll never open at the moment it matters.
Give a reason and you've handed over a problem to solve — and a solved problem becomes a yes. 'I can't this time' is harder to negotiate with than three sentences of explanation.
The follow-up message offering to help after all. It feels like decency and it's the guilt talking — it reverses the refusal and teaches the other person that waiting works.