I would be concerned about the buttercup being invasive.  It is not a native, as your research has no doubt informed you.  (Some natives are also invasive, although the preferred term is 'aggressive'.)  By comparison, the oxalis you are growing is slow to spread and easily removed. 

Although discouraged in lawns in NC, https://content.ces.ncsu.edu/dichondra, Carolina pony foot, Dichondra, is used as a substitute lawn, particularly in Florida, but does prefer moist conditions.  I found some growing in my garden, had to weed it out, but then put it in a place where it can run. 

You might also think about clover.