I used to manage a group of engineers exactly like your group. About 40 guys on four different levels and my job was manage everything NOT 9-5 Mon-Fri.
The shift they put you on is thought of as a nightmare by 80-90% of the population given that loved ones usually are not working on the weekends. And I also know that if you work those shifts on the weekend that your other weekend time is basically sleeping.
But the other 10-20% of the population absolutely LOVES that shift. You work 3 damn days a week and can do whatever you want for most of your week.
So the guys who worked for me that were successful at getting off their weekend shift came up and told me. I asked the others in the group if they wanted to switch.
The smart people sold their shift to the others. You could even sell it to people who aren't on your team that have the skills.... so sounds great right?
Well... the Mon.-Thurs. x10 shift was actually something I started at my company to help retention. This only went to my best and most tenured guys. I had guys stay on this shift and pass over promotion after promotion that would have made them go 9-5 x5.
So your best bet is to see if you can get someone to switch with you. Even going to your manager may not even be the best thing. If someone came to me and said this surely the other weekenders might hear and if they have more tenure than you or if I think they are better, I would let them switch first. But if you came to me with a 1 for 1 switch I would probably OK it given like positions.
In the end you got a permanent job. You are low man on the pole, probably asking for a premium position. You are also working a shift with little to no social game interaction with decision makers... If this is a deal breaker for you I would start looking elsewhere.