Will the prestige of my masters program affect my phd application choices?
I would recommend that you apply directly to doctoral programs and use this offer as a backup if nothing materializes. In mathematics and some other fields acceptance into doctoral programs with only a BS or BA is more or less the norm. Your vita looks pretty good and I think that many doctoral programs would be happy to have you. 

Most likely you would also be offered some sort of funding, though it would be as a TA or similar, though funded research isn't impossible. Places with doctoral programs in math in the US need a lot of TAs. 

Apply first and then consider your options. You might even be able to get some indication about the likelihood of your acceptance from a faculty member at a proposed university.

Also, in many doctoral programs in the US, you will have the chance to get a Master's along the way. Sometimes just by filling out a request after a certain amount of coursework. In my case a small thesis was also needed.