Just curious: I know you can get point-of-use water heaters (basically
just conventional water heaters with a very tiny tank, 3 gallons or
less, usually electric), on-demand tankless water heaters for the whole
house, and even instant hot water dispensers.

Is there any low-output reasonably low cost point-of-use water
re-heaters that migth solve the OPs problem? Something that could raise
the temp a few degrees for just a single faucet. Since it would only
run until the hotter water from the basement arrived, even electric
would probably cost little to run.

Not that I'd pay good money to solve what I see as a minor annoyance,
but the OP may want think otherwise.

-Kevin