What should I do with my Thai Elite visa when renewing my passport in Denmark?

OP seems to have got some idea. Let one of the moderators to re-open this post if you have specific information to be added. Thank you for your advice, everyone.

When you are coming back, arrange the EPA to assist you, they will take care of it. If you had renewed your passport in Thailand, usually your Embassy will give you a letter and with that letter, you go to the immigration and transfer your visa to the new passport, but since you are renewing it in your home country, I am not sure if your Embassy is aware of such a letter. I wouldn't worry too much as long as you get the EPA to assist you on arrival.

Usually, the old passport would be punched only in data pages. If you want to make sure, you need to let them know you still have a valid visa in your current (gonna be old) passport.

Then you will travel with both passports to Thailand. You will be stamped in your new passport based on the visa in your old passport, no problem.

Once back in Thailand, go to the Elite office and ask about the procedure again. If I remember right, Elite visa (it would be a sticker) is the only visa sticker that can be transferred to the new passport.

You need to talk with the Thailand Elite office for detail of that procedure.

In the worst case, you need to carry both passports until your TE gets expired.