I'd hazard a guess that as long as it's a Thai Insurance company because of the relaxation after COVID, they'd accept any company, but the internet is the place to find out for sure.
What's this thing some expats have! Why can't you do it yourself. Only two reasons, they aren't hard up enough to be bothered about paying an agent, or they don't have the required money, 800k and even if they do, why would anyone want that amount of money just sitting year in year out earning nothing.
I presume because you say the, not an immigration officer, that you entered Thailand 'the other day' ? the IO would confirm what you reckon 'the IO', said to everyone entering at Suvarnabhum? Or did you specifically ask an IO when you entered, or did you not enter at all and your just assuming 2 by land? Why would an IO tell anyone at the airport that they are only allowed 2 by land? unless they asked?
As long as your new passport starts before the date of expiration of your current one. What I did several times in Thailand, I applied for a new passport personally at the British Embassy. When they informed me I could go and collect it, I did and they cut the corner of my other passport, with my current visas in, then I went to any immigration office and they transferred visa to my new passport. Don't know today how you apply for a British passport in Thailand. I believe it has been farmed out to a private company. Don't know if that is true or not. I'd of thought for security reasons the Embassies, or consulates would be the only place that could process a new passport.