Assisted places at private school

You lose nothing with trying

A 4 year old with the reading age of a 14 year old?

The 'assisted place scheme' (which I suspect you might be thinking of, given your terminology) was abolished in 1997.

Each independent school will, if it wants to, offer scholarships (non-means tested awards for excellence in a particular field or all-rounder; can be large, small, or purely titular) and bursaries (means tested awards to those who could not afford to attend otherwise). In some schools you have to be scholarship level to be considered for a bursary; in others not.

How large bursaries can be depends on the school, how many are already in receipt of assistance and how healthy the underpinning fund. They go up to over 100% (though most don't).

You're going to have to ask. No use in getting your hopes up for a school where you'd need 75% fee reduction is the maximum they've ever awarded in their whole history is 50%.

You won't be laughed off the premises. Or at least not by any school that is worth going to. They get these sorts of enquiries all the time. Bursary applications should be handled competently and in confidence.

Assisted places was a government backed scheme which was abolished in 80s. You are looking for a bursary but for 100% the income level is often very low. Each school varies though so you need to meet the school bursar and ask what the level of income would be and how many 100% awards they make each intake. Sometimes they prefer to spread the pot thinner ie. 4 places @ 25% so that the number of beneficiaries is higher. Also ask what is covered and what is not. Uniform, extra curricular and trips can add considerably to the cost.

The wealthier and more well established the school the more likely it is that they maintain an extensive and generous bursary system.

It is much easier and more common for the "big name" schools to offer bursaries that cover the entire fee.

So ... it's a question of whether you are only prepared to consider this one school - and accept whatever they can offer - or if you're ready to search further afield.

No decent school would laugh at a parent in your circumstances. The best actively encourage less wealthy families with bright/talented children to apply regardless of income.

The best schools are very thorough with examining finances, quite rightly. They require both parents to work except for in exceptional circumstances, they look at all capital assets as well as the equity that can be released in the primary home. They also look at the finances of both parents in the case of separation or divorce, not merely the resident parent. Less good schools may not be so thorough but are also likely to be far less well endowed, so won't have anything like the same amount to distribute.

Each school has a different policy about bursaries and scholarships. It is possible that an able child may be awarded both a scholarship (tested according to ability) and a bursary but it is very rare for this to be 100%. Most schools would expect parents to pay something towards the fees.
Am not sure about Wales but in England the "assisted" places scheme was discontinued years ago.
You need to check this with the school (or name it as someone may know).
Also check how the state schools support high ability pupils. Some schools with average results still support high ability pupils well.