There are numerous online defelction calculators, the one I use is
here.
Using unknown wood/good condition gives: L/642
Using SYP or Douglas Fir gives: L/839
There are other calculators with permit a wider choice of lumber species.
Bump out to me usually implies cantilevered, but I'm assuming this isn't a 6 ft cantilever.
Maybe this is a diy/ignorance is bliss opinion, but I'd lay tile on that (with appropriate subfloor) without hesitation if that is the question. My parent's old kitchen had addition to the kitchen of comparable dimensions and 2x6 joists. I put down tile 10 years ago and it was fine when the sold it two years ago.