I'm considering gathering enough evidence and making an application for full custody without him knowing with supervised contact and divorce.
Do not do this. Make a log for your own records, but don't present said log as evidence because it is a waste of time and will not go down well in court. You can have men who are convicted of violence towards the mother and they get awarded contact. This will be seen as a parenting difference, not grounds for supervised contact. Even the legal stuff like not using a car seat. The most you will get is a court order saying he has to use a car seat - but even this is unlikely - there is a law saying that now and he isn't following it.
The relationship sounds like it's over, so yes go and see a solicitor, particularly to find out what your rights are WRT any finances, house etc, but don't wait around trying to amass evidence that he is a risk to her. One of the worst posts I ever read on here was from a mother who had always despaired at her husband's lax attitude towards safety and unfortunately this had resulted in an accident which injured her poor daughter terribly. I will never forget it. 
It is difficult to avoid unsupervised contact when you live together without being overly controlling - it is weirdly easier to do it when you're apart. He's not going to rush for 50/50 - he just won't. He wouldn't have time to gamble and play games. If I were you I would facilitate contact, ie, if he asks and comes up with a reasonable plan, then agree and make her available, but don't run around after him making arrangements or doing the organising for him, and do put totally reasonable boundaries in place e.g. no he can't just pop round when he feels like it with no prior arrangement. Most likely, he will rarely get round to doing the organising, and dine out on a sob story of "she never lets me see my daughter!". You do have to just become immune to this and tune it out. It is sad in a way but it's also a very low conflict way to go.
By this BTW I mean:
He texts you saying "When can I see DD?" (this is the usual low-effort request IME)
You DO NOT reply "How about Thursday afternoon?" - you reply something like "DD would love to see you. Let me know your proposed arrangements and we can discuss."
He will then either
- Go silent because making the arrangement is too hard. (No follow up from you needed).
- Start texting about irrelevant stuff like telling you how much he misses her or some other thing. Ignore this - or if it's bothering you or he is being persistent try "XH, we are separated. I only want to talk if it's something about DD which we both need to know."
- Reply something like "What works for you?" - trying to get you to do the work for him. Broken record this, reply with "Let me know what your plans are for seeing DD, and we can discuss."
- Reply with an actual plan, e.g. "I was thinking I could take her to see my mum on Sunday" or "I could come over in half an hour".
If you get answer 4, then decide whether it's reasonable for you or not and say
"OK, see you then"
or - "6pm doesn't work for me, as it's too late and gets her too excited to sleep. Earlier in the day would be better." (again, don't suggest days. You've given him info, he can suggest another day!)
or - "DD would love to see Nan as well. What time shall I drop her off?" (if you don't want him driving her) or even "What time shall we meet you there?" (if you don't want to leave her with him even with another adult there).
If DD gets upset about missing him between visits, don't text him and tell him she's upset or she wants to see him, or ask him when he is next seeing her. It is not your job to do his contact arrangements for him. Keep the rule of making her available, but doing none of the organising and running around (except perhaps in pursuit of avoiding him driving her).
Keep texts factual and minimal, never emotional.
Do not make suggestions for him. Do give him relevant information if he asks directly - e.g. he has established his plan to take her to the park at the end of your street on Monday for 1 hour, and he wants to know what time would be convenient to pick her up. Or he has established a plan to take her out for lunch, and is asking what food she likes - even then, minimal and factual. "Chicken nuggets" not "McDonalds' happy meal with chips and ketchup".
Imagine reading out all your own texts in court with "Your Honour" after them - this is a fantastic rule to stick to the principles of keeping DD available and never refusing contact, but also not chasing him to have contact. It's his call that way.
Good luck!