You can always reset the website to be able to choose a new template however you will lose your content. What I recommend doing is rebuilding out the site with the new template and the current content manually and then saving that as a team template and then resetting your site that way using the new team template you just created
So unless I am reading this wrong, You cannot:
#1 - Can Not transfer or in any way "move you site to a new template"
#2 - That you must build from new, i.e. "however you will lose your content"
#3 - "then saving that as a team template and then resetting your site" which sounds suspiciously like simply building a new template and eith Teams it or Use that one for hosting.
#4 - a well worded answer which does Not use the phrase 'you can't move your website to a new template. Well "OK" then.