I don’t see a judge ordering the kids to travel five hours, each way to see you every second weekend. It’s not reasonable to expect the mother to spend ten hours travelling 2.5 hours there, 2.5 hours back (times 2!) during the school term. The kids will be exhausted.
You have chosen to move away. It’s been your choice. I agree with the mother, you need to come back to visit with the kids, a reasonable distance. Ten hours driving is ridiculous and deprives the kids of engaging in their regular weekend activities.
Read the question before replying please. Your first response in particular is based on totally incorrect information.You made the decision, not the kids. Child support is irrelevant. Maybe you come & stay in an air B&B and not have the kids overnight. But 10 hours travel in a weekend during the school term is insane for the kids to have to do.
As for the kids being exhausted - they wouldn't be any more exhausted from a 5 hour drive than they would be from a 1 hour drive and 4 hours of watching TV. It's total nonsense.
As for courts - they will order that the kids spend time with both parents and what the OP has proposed (and mum rejected) is totally reasonable. There is no reason for a court to deny the OP's proposal.