I have a similar issue; I need to smooth my income because it jumps around, whereas bills don’t - they’re monthly. It would be nice if Monzo built this in to a native feature to be honest. You could even combine it with the rainy day fund in a way - you pay into a “buffer” which includes several months’ salary.
Back in 2018 I wrote a post about an idea that I think is quite related to this; you’d have 4 pots - discretionary, mandatory (i.e. bills), savings, and a buffer pot. The idea being that you assign your income to each, and rather than immediately transferring money to savings before you know if you’d need it and then have to transfer it back out, instead you’d deposit that in the buffer pot. If you went a bit over, you can pull from there. And only on your next pay cycle when you know for sure you can save it would you indeed save it. This way you can create more of a fortress around your savings pot(s), to discourage yourself from ever dipping in. Because as soon as you start dipping you can become numb to that action, and they’re not really savings anymore but more of a slush fund.