I also live on the Westside of LA in an area called Mar Vista that might fit your requirements - it’s not as expensive as MDR/SM/Venice but is on the “right” side of the 405 for you to ferry DC to rowing and rugby without being so close to the ocean you are permanently stuck in traffic looking for parking. Venice HS as previously mentioned has had an injection of cash and is doing better and better every year (though I would suggest this for thinking ahead for your 13yr old, not going into any American HS for the 16yr old) and there are a handful of decent MS in the area - Mark Twain is one that feeds into Venice and is a neighbourhood school that used to be terrible but has improved 1000 fold over the last few years. There are also various charter schools you can apply to, deadlines coming up very soon but as it wouldn’t be a normal transfer you’d prob just have to call them up and find out who has space once you know when you’re coming - look at New West, City Charter, Community Charter. Also look at LACES (you’d also have to live in the area for this or permit in but it’s much further east and north) as it’s more academic than a lot of places. There are several very fancy private schools on the Westside to consider for your 13yr old too: Windward, Wildwood, Crossroads and New Roads.
In terms of cost, a nice 4 bed single family home (detached house) in Mar Vista will set you back $6-10K a month. A lot more the further west you go but then you can get cheaper if you live in an apartment.
You’ll need two cars so factor that in - electric cars are actually great in LA because of the traffic and charging points everywhere, your DH will need a car for work unless he joins a rideshare and no way can you (assuming you won’t be working otherwise whoever will be managing your DC after school) can ferry kids from SM to MDR otherwise. But a good reason for at least one of them to be in walking or biking distance of school.
To give you some other ideas of costs so you can figure out how much he needs to negotiate - a salary of $200K/yr to live comfortably on the Westside is necessary especially if you want to make trips home, you can ask for those to be paid but I think most companies stopped doing that a while ago - doesn’t hurt to ask. His company should pay all patriation costs (they may well give you a budget) so you may as well ship everything over. And make sure medical care is excellent - you went the option to have HMO or PPO with Kaiser or Anthem Blue Cross as a starting point. Also check dental coverage, even the best plans only cover $2K at the most per family member per year.
The company should also pay for 60-90days rent (and organise putting you into a serviced apartment) when you arrive and cover the cost of a hire car while you find your feet, both these are pretty standard. And yes be guarantors on rental property after that. You will have to pay double down payment so be prepare for that - DH’s work gave him in advance so we could do this.
Luckily you can work on an L2 but you still need to apply for a work permit and SSN and it’s worth asking the company if after a certain amount of time they will sort out green cards for all of you, just gives you a huge amount of security.
Trying to think of other costs for you - groceries are much more expensive than the U.K., every after school activity is more expensive, and travel home is obviously a big budget item. Even phone contracts cost more than the U.K. but you don’t pay council tax (instead property owners have to pay property tax) and every school does major fundraising so budget roughly $1500/yr per child in public schools.
PM if you need any more help with budget/contract negotiation questions. Life here can be great but miserable if you have to scrimp and save especially if that means not being able to visit home or pay out of pocket for healthcare.
Just a note of caution - I would be very surprised if you’re here by the summer, all visa were suspended for ages and now there’s a backlog and it takes longer at the best of times without covid, just don’t pin your hopes on it happening quickly. At least it’s an L1, a H1B you’d be sitting on at least 12-18months
Also finally look into Redondo School district, I believe they have the IB and I think a school in Los Feliz does too - which could be an option if your DH was able to take DS to and from school every day. There is no way in hell you want to go from west to east twice a day for pick ups and drop offs!