I have a Shinseike growing next to a Bartlett and have already removed about 100 baby asian pears and maybe 20 Bartletts so the remaining ones will thrive. This does not make me an expert so I must say I don't know if the inedible Bradford will do the job. Almost any of the popular dwarfs sold at garden centers will work. I've seen them for under $30 at Lowe's, Kmart and Walmart. If a nearby neighbor has a pear it too will pollinate.

Bill, Thank you. I have space only for one pear tree. Will not be able to fit another one, even the dwarf. But my neighborhood is full of Bradford pears. Several dozens of them around, right next to my yard.
Olga

I'm guessing the bradford will pollinate your Asian pear. You'll know for sure in a year or two.

Most likely it won't. Bradford will most likely pollinate some European pears but not easily, and how much more if it were Asian pears. But Shinseiki is self-fertile. With or without the other pears it will set fruit. If there is another pollinating compatible pear with it, you will have problems thinning out the crop.

Waiting for 3+ years is a long time. Why not get a 2-in-1 Asain pear tree? It is a sure bet!