Tried with 47 uF, 100 uF, 220 uF accross the uC power input, no joy. One of the caps even made things worse (probably high ESR, it has been on the shelf for years). Also tried adding 3.3 nF + 15 pF in parallel to the transmitter's 100 nF bypass, no joy either.
As a sidenote, typical transmitter...