Catholic hospital doesn't allow tubes tied - annoyed

I'm expecting my 4th baby in spring. My 3rd was an unexpected c section with a very very long and difficult recovery especially with other toddlers to care for (my oldest is only 5). I'm approved for a vbac, and my dr said I could get an epidural and have the procedure done after I deliver... but the only hospital in my area that allows a VBAC, and is a high risk hospital, is Catholic and doesn't allow it. I'm not Catholic and neither is my Dr, but he said he can't do anything about it. So I either have to have another c section at the hospital where I delivered my other 3 kids, or go for the VBAC and get the tubes done at a separate facility after I recover (which is a pain in the ass, more money for me, but my Dr said is less invasive than the c section). If my first c section recovery was easy I'd just do another section, but it was pretty horrendous and I was unable to care for my other children for over 3 weeks (my husband and I own a business so he can't just take paid time off to take over for me, so I'd have to hire a nanny).

I guess I'm just pissed that the only high risk hospital in my county has the ability to dictate my health care decisions based on the religious beliefs of some. I'm not knocking those beliefs, but don't think that they should be able to dictate getting my tubes done if that's my choice. It's not like I'm having a late term abortion, it's my freaking 'old lady' tubes here lol. I'm 38 years old and having my 4th baby so I think I've well contributed to procreation here, but I have high risk pregnancies so should not have any more after this anyway, and hormonal birth control isn't recommended for me. My husband won't get a vasectomy (pisses me off too, that's a separate issue, but it's his body what can I do).

Anyway thanks for letting me vent and if anyone had their tubes done after delivery can you tell me how the recovery was?