Sorry, I am going to advise against that. Take the time to do things the right way and you will be far happier with the results.Bill, what is the
Feed a heavy duty large trash bag between the inner and outer hood pieces with the opening of the bag near the front of the hood - higher up. Attach a clear plastic tube to the foam tip and put the other end in the bag. Push the push into the bag and let it expand into the bag. That seems like it could work without taking the hood off. And if the opening of the bag is collected at the opening you had your finger in in one of your pictures you should be able to zip-tie it close to make sure that even if the foam crumbles it won't come out.
@Bluegill I assume you mean you don't want to remove the trim piece that is in the way, and not the actual hood itself, which is not necessary. Its super easy to take the trim off, especially if you order replacement clips in advance like the ones OP suggested. For anyone looking to DIY, you will also need to order the applicator gun for this foam. Use the foam SPARINGLY as a reinforcement and check the results before adding more. A little goes a long way.