If you are going to do that, why not install a NEMA 5-20R receptacle, that has two independent circuits and is backwards compatible, as well as being rated for 20A per plug?

The receptacle is the easy part, running the new circuit is the hard part.

Or you know, install a new 240V receptacle.

If I have to:

1) Run wire

2) Get a bigger breaker box

3) To do it legally, hire an electrician and maybe get a permit

Replacing the receptacle is like, <1% of what's involved there.