yellowcake64
I've read there are some interfaces whose ASIO drivers allow simultaneous playback - RME is one - but it seems the Tascam doesn't.
Yes, RME is... RME
It is a property of particular drivers what is allowed to work simultaneously. I do not have Tascam. In general:
* all interfaces allow multiple programs when Windows internal mixer is used to glue all streams together (MME, WDM)
* many interfaces allow simultaneously use ASIO (one program) and MME/WDM (all other programs), as long as the sample rate is the same
* rare interfaces allow simultaneous ASIO access from many programs
Finally, since you have 4x4, you can connect the output of your internal interface to the input of 4x4 and set internal interface as Windows audio output. That is what many people do to avoid problems (the fact some device support something does not mean it support that good or without extra limits).