I have a suggestion that will surely satisfy both sides: make Soviet Socialist Republics their own puppet type. A puppet with the SSR type would represent that the country is a full part of the Soviet Union while being a playable tag only for gameplay purposes.
Something like the Erevan Vilayet of the Ottoman Empire in Kaiserreich: the Ottoman Empire had many more vilayets, but devs wanted Armenia to be playable, so they gave it a tag but also an "Ottoman Vilayet" puppet type that clarifies that the tag only exists for gameplay purposes and lore-wise it's still part of the Ottoman Empire.
Giving SSRs their own puppet type would have the benefits of having special modifiers, so you can tune how they work better, and of giving room to add content for other SSRs in the future.