Hiya! I hope you're ok today. It sounds like you're starting to feel a wee bit better. Just keep celebrating the small wins at this stage. It sounds like you've been through similar phases before so you know how to do this even if it's a bigger hill to climb this time round.
You sound like a really caring, empathetic parent which is all a kid can hope for. Try not to sweat it with the kid, this is one of those brief times when you need to he a bit "selfish," or to use a better term realistic, about how much you have to give. Remember we put our oxygen masks on first, before kids, in an emergency and this is an emergency right now! It's ok to just focus on yourself and let the kiddo know you love them but it's mummy time right now.
You are doing everything right, you've been strong enough to ask for help, you're focused on trying to eat well and get out for those dog walks, you're working on the meds. Take each day as it comes, be patient and kind to yourself, celebrate each little win. You're really unwell right now and deserve some grace.
One thing I want to say at the risk of throwing another spanner in the world is have you ever been assessed for AuDHD? Your pattern of struggling with your mental health and 'burning out' during menopause along with a few other things you mention sound quite like my own journey. ... Just diagnosed with AuDHD at 46. It might be worth looking into when you have a bit more headspace to do so.
You got this.