Aaaah, the ex wives club of Mumsnet, it's great, isn't it?
OP - get the CSA sorted out so you know what you should be paying, based on your income and the time you have your child, and your living arrangements. Once that is sorted, pay that via whatever channel the CSA (CMA) says. Then do a full household budget, work out what else you can afford, even if only £5 a week, and pay that as well. Keep good records.
You should probably try and work more overtime. It's a bit confusing how the CSA takes more when you work overtime (so that you don't see the money) yet you say they have no record of you and you do not pay through attachment of earnings, plus pay directly to ex's bank. There is something odd happening there. You are misunderstanding something I think.
Make sure both you and your partner are claiming tax credits due to you. Your partner should keep chasing her ex/the CSA for maintenance payments.
Ignore your ex and your perceptions of what she spends money on, focus on your child and your current relationship.
As for how your partner feels - you have accepted now that 'love' might be too strong and I agree with that. I had a DSS for 7 years and I can't say I loved him. Maybe we all think of love as slightly different things, after all, there isn't really an agreed definition, is there? I would have been very upset had anything happened to him (and, of course, in 7 years things did happen to him and I was worried, was there for him, supported him - now we have split I miss him and try to keep in contact) and I guess you would say I cared about him. Maybe the way I feel is the way some people describe love, I have no idea.
But, it's too much pressure to ask her about it.
It sounds a bit as though having your DD over causes a load more work for your DP. This is unfair, you need to take the burden of that as it seems your DP takes the burden of 'running the house' the rest of the time, and then ramps it up every other weekend. You need to do half all the time, and then a bit more when your DD is there.
And, no, don;t try and do everything together. Sometimes you need one on one bonding, sometimes your DP does. Each of you with either child. Do you ever take your DSD out for a day and let your DP study? OK, you can't cook (learn, try YouTube) but you need to be providing the family meal at least a couple of times a week, even if it's just pizza, or fish fingers with oven chips, but you need to take your turn, let your DP go off ad study while you take full responsibility for the evening management. Do you ever do this? Can she rely on you to just be in charge now and then?
It sounds to me as if she is worn out, low from lack of money (though I doubt she has less than she did when you were living separately, did she think you would put more into the pot?) and then slightly resents the extra work your child causes.
It all sounds solvable to me, but you need to put your mind to it.