We actually also have version 4.2.2+ at my place of employment,
Well then the next point: Dont' use 4.x yet for production. In my opinion it is not ready for now.
You are right, the templates/themes after 2.06 are not compatible anymore. You can't use 2.06 templates in 2.7x, you can't use 2.7x templates/themes in 3.x and 4.x had another break there. I hope from 4.x onwards the themes/templates will be compatible.
But, while this argument is valid for a couple of months, I think 5.5 years is a lot of time to adapt themes/templates. I know it is hard if you have a lot of themes/templates, but I think security should be taken very serious, especially for a university (unfortunately we see a lot of very old installations especially at universities).
I know that Denis (sondages Pro) was supporting a fork of 2.06 LTS for a long time with bug fixes and security updates. Not sure how this is still updated.
github.com/SondagesPro/LimeSurvey-SondagesPro
As far as I can see, the last update was from February this year, so Denis seems to still work on it. Maybe he shows up and gives some more information.
Updating Themes/templates is not easy. It is far easier to adapt the standard themes to your needs. But one day you'll have to make the jump. From what I can see, you are from the US and it might not be as relevant for you. But having such an old tool still running for surveys will not be in your favor if there should be a data breach one day. Bases on the GDPR you are supposed to use state of the art software. So it will be difficult to argue that you did all you could do, to prevent such a data breach, because obviously the software was old, unpatched and with 3 newer major versions of course not "state of the art".
If 4.x would be "stable" (in what I personally would consider stable), it would be much easier. Right now you would have to adapt the 3.x templates/themes and then later the 4.x templates/themes. Or you just risk and go with 4.x directly, have to make the adaptations only once, but might run in some issues that are not working perfectly yet in 4.x.
I know, it isn't easy, but you must seriously consider on moving to a more recent version of LS (but please skip 2.7x, not worth the pain and also not supported anymore).