I don't have personal experience of Jigsaw, but I know of them, and I believe they are considered one of the leaders in terms of provision for early years. They have some heavyweight backers (eg Speaker of House of Commons, Bercow)
ABA has the most evidence behind it of any intervention. I think Jigsaw is informed by ABA, but offers something slightly different (lots of direct SALT for instance, and involving the parents to a much greater degree than usual). Their website is very informative. It is certainly head and shoulders above anything else I have heard of, or anything offered by my local authority.
If you are offered a place I would snap their hand off if I were you.
My DS is on an ABA programme and it is excellent. If you want your LA to fund an ABA programme you will most likely have to fund it yourself for 6 months or so (and demonstrate progress) whilst you take them to tribunal. You will not be able to specify a TA with ABA training. (Again, this would need to be directed by a tribunal.)
At tribunal you would have to demonstrate the provision the LA gives is inadequate/inappropriate and that the provision you want (ie ABA) is.
They will not fund willingly. Going to tribunal is a whole other subject in itself.
The LA Ed Psych is not allowed to specifically recommend a setting or type of provision. S/he will probably have to use a form of words such as "likely to benefit from specialist teaching" for special school placement. It is then down to an sen officer (with no specialist knowledge of SEN) to make the decision, after referral to a panel of LA experts. (The panel usually consists of Ed Psych, Special School head teacher, someone from social services etc). The panel can only make a recommendation, and it is the SEN officer that has the final say. Often SEN officers hide behind panel recommendations.
In my case our LA Ed psych was lovely, we had an open an honest discussion that started with her asking what do you want? She backed us in terms of special school / ABA which got her into trouble with the LA, but came in very useful at our tribunal.
Ed Psychs, though, are often "fettered", in the words of Brian Lamb, and only recommend intervention they know the local authority can provide, rather than what the child needs. I suggest start by asking what the Ed Psych thinks and take it from there.