Would using a Y hose connector to join rain barrel water with tap water be a good idea to boost the water pressure?
I wouldn't risk connecting the rain barrel to your house's plumbing because you don't want to risk getting non-potable water into your pipes.  This sort of arrangement might even be illegal where you live.

What if, instead, you just use a garden hose to add more water to the barrel as needed?  Then you don't need to worry about running out.  Or, use a quick connect fitting to make it really easy to disconnect your watering hose from barrel and connect it to another source.