It’s worth looking into, plenty of people at my DC’s school are lip service Catholics.
Check if there is a parish with a new priest nearby (newsletters are often online) or if you have a parish with a city, huge, transient population or several churches to avoid the Bridget-who-does-the-flowers issue. You can always say you are returning to the Faith after “resting” as you realise how important it is to you, especially in guiding your child.
Check the admissions guidelines carefully to see if you are in with a chance, if it is before 6 months, check the actual numbers of children admitted in the last 5 years in each category. If it’s a heavily oversubscribed school with only the top 1 or 2 categories you might not get in but if it’s not then you may.
Also be aware that the upswing of highly motivated parents (ie people like you, who can be bothered to check things like admissions criteria and do something about them) often impacts how successful a school is, rather than how good the teaching is. My own DC’s school gets tremendously good SATs but from Y4/5 a huge chunk of children are tutored heavily because they have very invested parents and we are in Bucks which is a Grammar County.
It’s competitive from the start in some Catholic (and Cof E too I believe) schools hence sometimes the excellent reputations.
Also bear in mind it is very difficult to recruit senior staff as head teachers need to be Catholic. We had an issue when our HT was unwell and there was a complete shortage of qualified candidates - I can’t see the Church being able to support schools the way they used to tbh.
Check too, things like voluntary contributions to the School/Church itself and uniform costs. Certainly our school tries to do a bit of backdoor selection that way and although one is voluntary, there are lots of charitable collections throughout the year for Catholic charities you might not want to support either.
As an aside, the whole experience has put my children off Mass entirely, which is something of a shame but there we go.
Hope that helps a bit.