(The only time this should be a problem is if your airport requires a call for engine start, but that's pretty rare).
Not that rare at major controlled airports, including Rotterdam, where I fly from. It serves mostly as a checkpoint for ATC that your flight plan is in the system.
But it's no problem whatsoever wrt. avionics.
- Master on
- Avionics on
- Listen to ATIS
- Request startup
- Avionics off
- Alternator on
- Strobes on; fuel pump on
- Start engine; post-start engine checks
- Avionics on
- Request taxi
ATC, at least over here, expects you to have your radios off at the moment you're starting the engine. They know you'll get back to them to request taxi, and if they have anything to add, they'll do it then.
The only "issue" that you might have is if you have a fancy nav/com/gps box such as a GNS430 which requires a boot procedure and is only ready to transmit, or change frequencies, after 10 seconds of boot process and four button presses. For that reason I always request startup using the "old" instant-on radio that's installed as COM2.