Whether they'd offer 6 months net pay would really depend how they viewed the situation, and whether they thought there was a risk of you successfully claiming constructive dismissal or generally kicking up a stink (grievance etc). They may well go for 6. However, as a rule of thumb I'd expect they'd go for at least notice (I'd expect 3 months?) which would either be paid in lieu, or as garden leave, plus perhaps another month ex gratia payment (which would be tax free).
What you have to remember is that if there is a genuine capability issue (not saying there is but if that's how they are perceiving it, or if you feel so stressed / unwell now that you it could develop into that), then it would become an HR headache for them having to go through the motions with you and ensure they are doing capability procedures correctly and fairly - employers often cock up and managers are often too busy to want to deal with it. If you raised a grievance they'd also hate having to deal with that. This all gives you leverage as they would rather have a clean break and not risk any claims / hassle.
Definitely speak to an employment lawyer - most offer a free initial call and they will be happy to talk through your options. If you go down this route they can help you make a settlement approach in writing (i.e. draft it for you to send), or help you ask for a protected conversation to discuss the situation in a meeting (they wouldn't attend that but can give you pointers as to what to say). This may also in turn lead to a settlement offer.
4 managers since you came back is chaotic - no wonder you're stressed - especially when managing the general single mum juggle and experiencing confidence eroding management styles. Excluding from pay reviews sounds off too. Definitely enough to work with here for a settlement approach. If you have any long term MH issues and feel unsupported that would also be very relevant and would likely help get you a settlement.
I know some have said speak to them about it first and see if that helps. That might work. But I in my experience solicitors know what's going on and sometimes for various reasons get a set idea about someone that they then won't shift. I guess it also depends whether you would ideally prefer to stay there long term, if this is a blip that could be sorted out. From what you've said it sounds like maybe this firm is all over the place, and so I'm not sure that's the case.
Sounds to me like you've been unlucky, but also perhaps you would find things easier to manage generally elsewhere, either another firm or in house. For example, at least in house, even if you had complex new work you were less familiar with you wouldn't have the constant pressure of billing, time recording, bringing in work, BD initiatives etc, and you could dedicate more time to brushing up and learning. One thing I would check though is what the internal support networks are like - there are often buddying schemes / online networks in bigger organisations, or you may have a decent supervisor allocated.
As for worrying about sick leave - I wouldn't. Some employers just go by the employment reference (which will 95% of the time only refer to dates of employment and role) so would never even ask or know about it. Bigger firms or organisations may well ask you to disclose any sickness absences within say the past 12 months on a health questionnaire, but this should only be sent out after you have received a job offer. You would need to disclose it but just be brief but open about the reason. If it is linked to a disability (i.e. long term MH issue) then they'd have to discount that as disability related sick leave anyway (unless it was a shocking amount e.g. years). I have long term health conditions all of which I disclosed, and the jobs that asked about it were not put off by it at all.
One tip - when getting signed off sick by your GP it would be better for them to include 'anxiety' or 'depression' rather than just 'stress' - as those are more likely to be considered disabilities than stress (they wouldn't be disabilities unless they were long term and substantial conditions, but a new employer would prob be more wary about raising any issues with it as they sound more legit than just 'stress').
If you only had a couple of months off due to anxiety, then I'd be really surprised if you had job offers withdrawn on that basis. It'd help if you can explain a narrative for why you became anxious for that set time about an issue, and how it has now resolved and won't be likely to come back.
I'd recommend speaking to an employment lawyer before you take your next step. Perhaps try a couple to see if the advice is broadly similar. They would get paid for advising on your settlement agreement (if a deal is reached) and are paid recently for this (by your employer) and it is relatively easy work, so you should find plenty of takers.
Good luck OP, this phase will pass!