I thought you told us that the Institution has to sign the letter, not your former boss. Remember the grant that pays your salary was issued to your institution, not your mentor, so the administrator in your grants managment office should sign the letter. The letter is just to request a change of mentor, correct?

It is typical for equipment to transfer with a PI, as long as the research grant that paid for it is also transferring institutions (at least in the biomedical sciences). The paperwork isn't complicated, its part of the paperwork for transferring a grant, which the PIs new institution gladly will prepare.

I think you need to stop feeling sorry for yourself. PIs move, he offered you a job, you didn't want it, so you need to live with your choices.