OP, your DP is in absolutely no position to be making any sort of demands upon his ex. I am not even going to comment on the private email and webcam suggestion as others have already been pretty succint on those suggestions.
Your DP is simply not trying hard enough. I did family law when I was training and I dealt with a lot of men like him - lots of whining about their rights and all sorts of wants and demands, but not much willingness to put themselves out at all. I also know someone like this personally and have the greatest difficult in not banging his head against the wall whenever he starts moaning about not seeing his kids after not bothering with them for weeks.
I am extremely dubious about your explanation as to why contact is in a contact centre - unless things have changed a lot since I did family law, contact centres are a last resort. But in any event, if contact is through a centre then the court/centre can be involved in discussions about a change of days and times to allow the fortnightly contact to continue. Telling the ex that it is "tough" is unbelievable. Why she hasn't turned round and said 'well you're not seeing them at all - tough' I don't know - perhaps she actually prioritises her children over her feelings about her ex.
Why is your ex not paying maintenance? Even if it is a minimal amount? Has the ex been to the CSA? I have never believed that contact should be dependant upon maintenance, but it does rather remove the moral high ground when your DP is not paying a penny towards these children.
Your DP needs to sort himself out - and you need to stop playing down his behaviour as "mistakes". The situation is entirely of his own making. He left his partner for another woman, removed himself from the birth certificate of one child, does not pay maintenance, got drunk in charge of the children, got stopped over the limit with them in the car, attacked his ex's partner, got banned from driving which contributed to the supervised contact he had slipping to monthly at which point he told his ex "tough".
The ex must be a saint to even entertain anything not directly ordered by a court. He needs to sort himself out sharpish otherwise his children will grow up with no relationship with him and no interest in him. I speak as someone who is only just establishing a very basic relationship with her father after 5 years of decreasing contact followed by another 17 of no contact.
A good starting point would be to train yourselves to stop blaming the ex for everything.