Nothing with exsi will been cheap there is basically .0001% chance of you finding this

Whoever told you all VPS places prohibit using their servers as a data saving/backup device is an idiot and should be ignored.

You can migrate emails to a new provider , I have done it for companies before , find yourself someone you trust to do the work for you,

$48 is expensive? lol

Directadmin + crossbox is a nice combo I use neither are free tho