Ok I know this is hard, but I know what I am talking about OP. Get completely out of the mindset you are in, worrying/obsessing about your ex and the rights/wrongs of the situation will drive you mad. Your focus should be 100% on the children. I know this will be a tough thing to hear and I expect you to push back on this, but worrying about what your ex does/doesn't do is obstructing your thinking.
Focus on your kids, and realize that your ex isn't calling all of the shots, all that is happening is you are letting her. I wouldn't actually advise court if you can possibly avoid it, but goto a solicitor and just have a meeting outlining how much you see the children and what you would ideally like to have happen, and see what they advise. One of the things they will point you at is mediation.
Save some money though in case court becomes a necessity, but write in a clear and concise manner to your ex (in fact you want to start relying on written communication so you have it to refer to) that whilst you are grateful you see the kids as much as you do you are keen to start having them more and more and write down a reasonable proposal for shared care (and get out of your head of 50/50 being some sort of holy grail of fairness!), and invite her to respond with what she thinks is fair. Do not make threats, do not threaten with court, just be cordial and respectful. If you just keep getting flat out denial or refusal to engage write again, without trying to press her buttons that you would be open to discussing this through mediation.
Keep logs of how much you see your kids, maintain an impeccable demeanour when dealing with your ex AT ALL TIMES which isn't to say let her walk all over you, but do not take the bait, do not get drawn into arguments, just state your position clearly and leave it at that.
If after giving your ex every possible opportunity to co-parent with you effectively goto your solicitor and take their advice on how to proceed. I know you hear horror stories about courts being biased towards men, but I have been through this process and from my perspective nothing could be further from the truth. Where tonnes of people go wrong is they are combative, and argumentative with thier exes and this makes the whole endeavour a massive mess to untangle.
Courts will always try to make decisions based on the best interests of the children, and are not trying to take one side or another in gender wars. I've been through it and actually this common misconception may work to your kids benefit, if you go in composed, polite, respectful and there can be no doubt in any judges mind that your ONLY concern is for your children wheras your ex goes in combative and argumentative she will be shooting herself in the foot severely.
But only goto court if you have no choice, as you'll have to salvage a co-parenting environment with someone even MORE hostile if you go to court and she feels she loses. This might protect you and your legal position, and sometimes you have no other recourse, but psychologically it is not ideal for your children. Best of Luck!