An epoxy may be your best choice for a problem such as this. Most epoxies do not react with foam products. On the negative side, cold temperatures are not conducive to good bonds, which may result in days or weeks of cure time if the refrigerator is in use.
Even though you reference not using a CA type of glue, you'd be applying it to the plastic surfaces and not to the foam. You would have no strength by applying adhesive to the foam regardless of your selection.
The above video shows a series of tests performed on plastic being attached to steel, using various types of epoxies and a couple of CA glues. The winner, in my opinion, is the Loctite Gel Control Super Glue. I've used this product and it's easy to apply exactly where needed.
You would be able to avoid applying it to the foam and even a bit of overflow would cause no problem. If you want to ensure no contact, consider to apply a layer of ordinary wood (or paper) glue aka PVA glue to the foam (only) to seal the surface, but I think that's overkill.
Another super glue type adhesive with which I've had success is DAP Rapid Fuse, which is a bit more expensive, but good product often costs more.
If you absolutely have to avoid CA glues, it appears from the video that the old standby, JB Weld may have sufficient strength. I can attest that it's strong stuff and has worked well for me in the past.