The problem is the Culcheth registers have many missing pages.
You can view the book of the registers on MyHeritage. (I can't see if the registers are on familysearch, as the 'search' and 'search catalogue' facilities are not working - they have done a big 'update' - HA!)
It looks like whoever was the incumbrent there in the mid 1600s did not do a great job at keeping up the registers. Perhaps there was
no incumbrent for some of the time, (it has been known).
Between about 1654 and 1681 there are no marriage entries at all, and some of those years there are no baptisms or burials; some years there are just one or two - usually for the family which looks as though they were the Lord of the Manor, (Holcroft) - perhaps he filled in the register himself.
It looks as though these are the years in which the other children will have been born.
Do you know for a fact that the name of the mother of the first two children was Jane?
I can see no mother's name on their baptisms.
'Henry son of Willem Bete 4 febrery 1665' - as written.
Although you have found a marriage for a William 'Baate' to a Jane Hussey in Warrington in 1665, this may not be the same William Bate.
Have you searched for other children around Warrington for this William Bate and Jane?
If the name of the wife in the will is Margaret, perhaps that had been his one and only wife, and their marriage is one that is missing from the Culcheth registers.
No burial for a Jane Bate, but again missing years at the pertinent dates.
A Margaret Bate was buried 5th June 1712 which is probably her, although again nothing else is written in the register.
As the Chucheth registers are a poor source of information, other avenues could be helpful
Have you found the parents of Jame Hussey?
Did her father leave a will? There may be some mention of Bate grandchildren.
There are baptisms to other Bate families in the Chucheth registers - I saw Roger, Robert, Gilbert and John, but there may be others. You should check.
Have you followed these other Bate lines? Did any of then leave will? Once again your Bate children may be mentioned.
I have found the wills of bachelor uncles and spinster aunts to be particularly useful, as, having no children themseves, they tended to leave items to nephews and neices and wives of their brothers and sisters
I did see the surname Bate in the Culcheth registers in the early 1600s.
It looks as though they have been there for some time.
I see 'Wm Bate junior,' along with 'James Batte' and 'Hamlet Batte' (I saw a couple of baptisms with him as the father), are listed in the 1664 Hearth Tax at Culcheth.
'Henry Baite' is at 'Croft' - another place that turns up in the Culcheth registers - it must have been very close). 'Margery Baite' also at Croft, John, James, Robert, Cicely ................
https://gams.uni-graz.at/o:htx.lnc5