Our Lady of Compassion R.C. Church was founded 1796. Baptism & burial registers from 1796 and marriages from 1797 are at Lancashire Record Office, according to GEN UKI. There would have been a Catholic mission long before then, so Catholics in Formby would have been baptised and perhaps married by priests without the ceremonies being recorded. Formby didn't really take to the Reformation. I came across a list of priests but didn't take a note.
I've had a look at St. Peter's registers on LAN OPC. There were several unnamed children of William Formby and of John and Joseph Rymmer in the baptism register of 1760s. Perhaps they regarded it as a minor act of defiance not to give the names of their children to the vicar. Have you come across baptisms of William Rymmer, Jan 1764 and James, Oct. 1765, father of both was John. According to a note on LAN OPC the word "rarity" was written after surname. This may be a place, but I wondered if it was a comment on John Rymmer's infrequent visits to church. There were a lot of Rymmers in Formby as you'll notice.
For each of the years 1797-1800 there's a list of Catholic baptisms (or births) after the C. of E. ones. Not for 1801, '02, '03 etc. Puzzling that they started again 1808-1810.
There's a question on this topic on Liverpool & South West Lancashire FHS forum. Thread name is Parish Register - St. Peter's, Formby. The questioner found his Catholic ancestor in the register in 1780s.
https://www.liverpool-genealogy.org.uk/phpBB3/vewtopic.php?f=2&t=14993St. Michael, Kirkham, Lancs. has lists of Births of Children of Recusants for several years in the 1st decade and later in 18thC. The transcription on LAN OPC for the baptism register 1776-1800 has this note at the top of each run of years: "This register contains 3 batches of Catholic/Recusant children. It is unclear whether the 2nd batch is births or baptisms. For the purpose of this transcription they have been entered as baptisms." It was so unclear I couldn't find them. I didn't delve into St. Michaels' 1760s register.
My Lytham Catholic families and their associates took up a lot of space in St. Cuthbert's C. of E. Baptism register for most of the 18th century until the start of 19th. Lytham, like Formby had a high proportion of inhabitants who kept to the old faith. The format at St. Cuthbert's was to record a Catholic child as a "Birth" in the baptism register. It wasn't consistent though. I've noticed lot of Catholic children recorded as "Baptism". 2 versions of the register from 1750s survive. Some entries which were "Births" in one were "Baptisms" in the other. Recording of Births accounted for between 1/3 and 1/2 of entries for most of 1760s. "Pap" was written beneath birth entries for a few years in 1770s. The earliest birth entries I saw in St. Cuthbert's register was middle of 1698. (after Protestant William of Orange took the British Crown from Catholic King James and James tried to get it back; cue rising in Scotland and war in Ireland.) There were 9 children; I recognised 4 of them immediately as R.C. So then the listing of names was to control a dissident element in the land.
The 1760s recordings may have been early preparation for the Papists' Returns. Or maybe some clergy were busybodies.
The tax laws I mentioned in my previous post were:
1.Marriage Duty/Registration Tax, abolished 1706.
2. Stamp Duty Tax 1783-1794.
The 1st taxed births (not baptisms), marriages and burials, childless widowers and bachelors over 25. The rate was 2 shillings (10 modern pence) to register a birth, 2 shillings and sixpence, half-a-crown in old money, for a marriage and 4 shillings for a burial. Some clergy compiled lists of residents. Some parents tried to avoid the birth tax by not registering births of their children. A seeming increase in twins in the years immediately after 1706 may have been their tax-evading parents catching up with a back-log of baptisms.
Stamp Duty Tax was similar to the earlier tax. Again, some parents didn't register births.
See The Gen Guide - Marriage Duty Act/Registration Tax (Parish Records & Tax Records)
https://www.genguide.co.uk/source/marriage-duty-actregistration-tax-parish-records-amp-tax-records/184I had a quick read of "Formby Families of Formby", it's only 3 pages. tsgf.pbworks.com/f/Formby.pdf