Could you plant it towards the east dripline of the big tree so that it gets more morning sun?
There's gotta be more than two spots available :)

Every azalea variety has its own requirements. The problem with too much sun is some azaleas may get lace bugs. Leaves infested with lace bugs have white spots on the top but on the under side of the leaf you find the bugs which have transparent wings and leave varnish-like spots of frass. Some varieties are resistant to lace bug and do well in more sunny locations.

In too much shade, an azalea with thrive and have green leaves, but may not set flower buds. So it is best to leave it where you want it and see if you have any problems. Then move it if you do.

Here is a link that might be useful: Sun and Shade for Azaleas