Using Migration Assistant

If you have extra users already in place on the new Mac, then you can go to System Settings > Users & Groups to delete them and their home folders to free up space.

Delete a user or group on Mac - Apple Support

If these extra users are on the TM backup drive, then when you run MA, you must only select the user account you wish to migrate to the new Mac. This may involve clicking the disclosure arrow next to a selection choice when you are prompted to "Select the information to transfer".

Transfer to a new Mac with Migration Assistant - Apple Support

+1 on den.thed's approach. You cannot put 1 gallon of milk in a 1 oz. container which is what you are attempting. Your MM should have the same amount of storage as your old iMac. When you get the replacement MacMini please follow the steps in Setup Assistant and Migration Assistant for a smooth and trouble free migration.

You can not properly migrate to a new Mac that has a smaller storage drive.

I recommend that you return that Mac mini and buy or order one with a 512GB or 1T storage drive.

Buy Mac mini - Apple

Do not transfer large directories like photos and put them on an attached drive to use as storage to work along with your smaller internal drive.