Trying to arrange private arrangement but (soon to be) ex wife has gone behind my back and submitted a CMS claim

Hi, 

I'm new to the forum. I'm an honest, caring Dad who just wants to do the right thing for his children (x2 aged 9 & 10). Some high level background context before I go into my query:

• A few months ago my wife informed me that she wanted to move out and separate in order to have some space, assuring me that no other interest in men, but just seeking space to think and work things out.
• I supported her decision and went above and beyond to help her get a rental, providing up-front rent, £thousands in furniture etc.
• When she originally moved out, we did not have a long term agreement on childcare arrangements but she basically set the stall of having the children most of the time (i.e. only allowing them to stay over with me in the marital home for on average 1 day a week plus additional days for holiday and birthdays). I was kind about this as her having the girls with her would reduce stress and help her think/have the space, but on the basis that we would agree after a few weeks/month or so a more permanent arrangement.
• A month or so after she moved out, I discovered that she'd been having a secret affair since the beginning of the year that had turned physical (before she actually moved out, and whilst she pleaded with me to support getting a temporary house for her and the girls). 
• I decided that I wasn't happy and submitted a divorce application and informed her of this. I did not mention the affair as I did not want to be gaslighted.
• Roll on a few weeks, and she and the man were confronted by his wife and everything came out in the open, albeit she still denies having a relationship with him...even though he admitted it to his wife and there is conclusive evidence.
• I have since been trying to discuss childcare arrangements as I would like them to spend equal time overnight with both parents.  I never agreed to the current interim arrangements in context of the updated circumstances! I have the ability to do school runs, etc. and have the same if not a much better position compared to the mother.
• Discussions and conversations on an arrangement have not been fruitful and she is not currently prepared to the change the arrangement to 50/50 shared care (i.e. 7 nights each over a 2 week period). 
• She has said she will now only talk via mediation. I took this on board and submitted a referral for a mediation service shortly afterwards and I'm hoping it will help us to come to the middle ground and for her to realise that my request for 50/50 is reasonable. My solicitor believes a court would likely agree with it based on circumstances/positions of both parents and the children.

She has now gone behind my back and submitted a CMS claim without telling me (in fact it was submitted just before we had discussions about making a private arrangement). Based on the letter I received today (dated 30th May), the calculation indicates that she has put down that I never have the children overnight, which is a complete falsification. 

Is there anyone here who has experienced a similar situation (not necessarily the background context), with a false CMS claim? I have to respond within 14 days of the 30th and looking for some advice or guidance on how best to do this. Ideally I would like to get a private arrangement agreed, but that could take weeks through mediation, or worst case, court if it's unsuccessful.  Am I right to think that I'm a bit screwed in having to pay CMS based on the current non-permanent arrangement or even worse, the full amount that she's falsified on her application?

Thanks in advance - and if anything is not clear, please ask and I will clarify!

Another caring Dad.

p.s. it's great to know this forum exists!