Seeing how most employers deliberately constrain their hiring prospects by choosing very selective and specific venues to place job ads, I don't see why you couldn't do the same. Just advertise for the job with local LGBT organizations, on sites like this (is there, by any chance a Help Wanted Forum on this site? (hint)), or by word of mouth through any crossdressing contacts you may have. I find that 90% of the time smaller companies have the best luck finding the best employees by going through word of mouth and (EEEK!) networking. Sure, you'd love to have the greatest, best, doctorate educated, management ninja Forbes couldn't interview because they couldn't find him/her, but you also want someone who will mesh with the company culture and it sounds to me like the company culture is crossdressing friendly.
If I had any construction experience beyond home improvement work and tile laying, I'd be thrilled to relocate to AZ.
And I think it's time to open a Help Wanted forum here.