The book "Lighthouses of Liverpool Bay" by John and Diane Robinson has quite a bit of information about the Hoylake lifeboat in chapter 3 (which is mostly about the Hoylake lighthouses). You can usually find a copy on ebay or abebooks. John and Diane Robinson also compiled the information about the lighthouse keepers of Liverpool Bay which is on the ALK website:
As you say, William Bird kept Hoylake Lower from 1819 until his retirement in 1855. Another Bird, Henry, kept Hoylake Lower from 1860 until his death in 1868. If they were related, I have not found the connection (but I haven't really looked). Nor am I aware of any connection between the John Bird that you mention and Liverpool's lighthouse service. Bird was a common name in North Wirral, especially amongst fisherman in Hoylake.
John and Diane write on page 73 "Joseph Bennett seems to have become less active, perhaps through old age, in the years following the Earl Moira incident. By the end of 1825 the whole of the lifeboat service, locally, had become moribund, so a special committee was set up to deal with the problem. After a year a full report was produced. This resulted in a new and improved lifeboat service that operated from all points around Liverpool Bay. Joseph Bennett remained keeper of the Upper Lighthouse until he died in 1828." They don't give a source for this particular statement. But it does suggest that the relationship between the lighthouse and lifeboat probably altered around 1825.
The Merseyside Maritime Museum Archives hold the extensive Mersey Docks and Harbour Board collection (which is incompletely catalogued). Now, the MDHB wasn't established until 1858, so its primary records won't be contemporaneous with the period you're interested in. Nonetheless there may well be information in those archives that touches on your questions.
Prior to 1858, the lighthouses (and I presume, lifeboats) were operated by the city of Liverpool. So you might have some joy from their records; Liverpool Central Library would be a sensible place to start.