Can XPS foam support the weight of a bath?

It depends a great deal on the particular foam. XPS is available in many compressive strengths, though probably you only get one option at mass market home stores. That option would appear to be 25 Pounds per Square Inch (PSI), based on some quick product research. So a 500 lb tub+occupant would be held up by 20 square inches of that foam, if the load was applied evenly (i.e. the foam must fit the bottom of the tub well, or you must use an interface material that fills the gap nicely - I'm personally fond of the old-fashioned but highly effective plaster bed for this part.)

If you put a curved bottom on flat foam without fitting the foam or using an interface, expect to have high pressure at the point of contact and potential movement.

From a better class of supplier you can get very high compressive strength versions... 40, 60 and 100 PSI appear to be commonly available, based on more quick research. But the fact that the bottom of a tub is far more than 20 square inches means you probably don't need them - if you spread the load onto the foam evenly.

I (and many other people) have 25PSI XPS foam with a concrete slab floor sitting on top of it - it can take loads just fine if they are within spec and spread so there is not excessive point loading. I've driven a 7000 lb tractor on that slab floor, and nothing moved appreciably/noticeably. The load could be held by just under 2 square feet of foam, and is in fact spread across many more square feet.

Oh - the primary effect of sheet thickness (assuming you have a floor under this design project to which the load is ultimately being transmitted) is the potential for gaps between sheets if they are not assembled with care. If there's no floor and you want to transmit the load to the end frames it's a whole other ballgame and you might want to call in engineering help.