Cold, raw, white-hot metal. Mixture of Track, ATB, Hybrid with aftermarket parts. a bike for 1%er. SNAFU

Will higher teeth on freewheel (eg 22t) reduce rolling resistance as much as the lower tooth chainring?

I don't think this is a relevant argument (for you and this bike), especially since budgeting is a (big) theme of this project (and not speed/efficiency). So I'd go for the cheaper solution.

I think you're correct in that less teeth cause less resistance, but firstly you won't be able to notice this and and secondly less teeth wear down the cogs/chainring faster.

If you're talking about the gear ratio then yes indeed, it doesn't matter if you use a smaller chainring or a bigger cog/freewheel; the ratio between the two changes in a way that the pedaling will become easier using both methods.