What is the process of this move internally? Do they even allow it?
Yes it is allowed
It's not uncommon. You can either interview again or find a team to "sponsor" your conversion by giving you SWE projects with which you can prove to have SWE skills and convert in the next cycle. By SWE skills I mean performing well in working on product features, both SWEs and SETIs need to have the same skill levels regarding coding and systems design, the only difference is the focus (infrastructure vs product features). They have the same salaries and ladders.
Not easy however