I can't tell if that foam gasket outside the surround is actually part of the surround or can be removed, or how well its bonded to the basket to use in a structural capacity, but next question would be "are you looking to mount traditionally, or mount from behind?"
I say that because if mounting traditionally, and opting for silicone adhesive - cutting the tabs off look to be no problem, no risk.
If mounting from behind (like to a trim ring then securing the trim ring to your baffle), I'd want to know if that gasket were secure enough to use to adhere to, or if I'd want to remove it, then adhere to the basket directly.
Either way, you might be able to drill tiny holes right where that gasket is, and use screws - it might be easier to drill if you don't have the gasket there. If it's attached to the surround you really need to be careful, since a spinning bit could catch and twist the whole surround, ripping or mangling it - so extra caution, consideration, and care, there.
If attached I'd hesitate to cut a circle inside the gasket to separate it from the surround, since that would reduce the surround's area of adherence, maybe it would let go over time requiring re-gluing.
But in theory - cutting those tabs off - that itself will be harmless if you have a good way to do it that doesn't cause too much heat - as that could also weaken the surround's adhesive bond.
If you can be super careful, I think there's lots of options for you.
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