Yeah from what I read, a 4 wire dryer is apt to be a combo of 220 and 110 and the latter requires a neutral(white) wire that 3 wire 220 doesn’t have.
I had a thought of what I would probably do. Where there’s a dryer, there’s usually a washer which plugs into a 110 outlet. That 110 outlet has a white neutral wire going to it.
I my case, I’d probably end up with a white running across the outside of the wall from 110 outlet to 220 outlet tucked under each outlet cover and I would say “for now” and call it temporary but...
Of course if both outlets are in the same wall cavity between studs, it would be easy to do behind the wall, if it’s a even a wood frame wall.
Alternatively, something could maybe be done in the attic space right above the washer/dryer. Add a junction box in each circuit for tapping into one for the other. Even if romex, you’re not really supposed to have exposed wire nuts in an attic or wall cavity. Supposed to do it in a box with romex connectors that clamp down on romex to prevent pulling on the wire nut connection.
In any case, I would personally try to avoid running new 4 wire romex from the breaker box all the way to the dryer. I would find me a white wire somewhere nearby to tap into. 110 outlet. 110 light etc.
If there is indeed 110 items on the dryer, it sounds like very low power items. LED display takes nothing and if it has a light somewhere, it’s not a 100 watt bulb. Maybe 15 watt bulb like a fridge/oven or it might even by LED.
I wouldn’t have a problem with tapping into a neutral somewhere nearby. But that’s just me.
I would try to find out what in the dryer takes 110 and also consider what kind of load the 110 circuit I’m tapping into has and ballpark the total amps. All white/neutral wires go to the same terminal block in the breaker box so technically, they’re all a single circuit with a lot of branches.
Heck, the manufacturer is already tapping into one 110 leg of the 220 instead of having a 110 cord plus a 220 cord.
Hopefully you can find an electrician with creative common sense.