I suppose it could be wired that way. I had it permanently wired myself that way on my Les Paul...but I guess anything can be done, although I'm not entirely sure if normal push-pull switches allow for that kind of swapping of contacts. A tech will know better.

On a coil tap...how was it again...you basically only need to switch the other coil out of the circuit, select if the hot is connected to 1st coil or direct to 2nd?...uh...anyway, that a simple switch can do, but with split singles, hm...you'd have to switch the 1st coil return to ground AND simultaneously switch the other coil's send to hot? (I'm just thinking how it must be done in my head, might get it wrong too but you get the idea anyway)...so yeah, if I understand correctly all that is needed is dual switch that can do two separate switches in one movement.

I do think those push-pulls can do that, they tend to have so many contact studs on 'em I'd be a bit surprised actually if they couldn't do it.
I bet you could also add another switch to have both options available.

I've only done such soldering mods a few times without clear instructions, trial-and error and thinking hard method, so ask a tech...sorry I can't give you a definite answer, but I'd guess it's no-brainer for a good tech...but I do very much prefer the dual single mode to simple tap.