I don't think there should be a problem with trying the eggbutt snaffle. If Happy shows you that she doesn't like it, you can always start swapping the bit over then.
I would try the bridle on her while she's just tied up in the yard and you're grooming her, so she can get used to the different feel of the bit without the added pressure of being ridden at the same time.
Then I would school just at walk for 5-10 minutes. Start by going round the school a few times on both reins and then move onto some 20 metre circles on both reins. That way, she should get used gradually to the feel of the bit and its different action in her mouth. After that, start on some walk-halt-walk transitions, and then walk-trot-walk. If she's okay with all that, keep schooling as normal.
I wouldn't worry too much about it being a stronger bit - so long as you keep a nice, soft contact you shouldn't cause her any discomfort