Would the ' inn' not have been part of the Priory, which had been dissolved around 1536. Weren't they a hostelry too?
Ralph Wolsey leased it with immediate effect. http://www.british-history.ac.uk/report.aspx?compid=39974
(paragraph 8 )
The priory was included in the list of monasteries worth less than £200 a year and was liable for dissolution under the terms of the Act of 1536. (fn. 62) It was probably dissolved in May or June 1536 as the prior was awarded an annual pension of £12 at the beginning of July; no deed of surrender or inventory has survived. (fn. 63) In the following year the former prior and four monks were dispensed to hold benefices with a complete change of habit. (fn. 64) The site of the priory was leased immediately to Ralph Worsley, a member of the royal household, (fn. 65) and in 1545 Worsley purchased the site and most of the priory's lands in Cheshire for £568 11s. 6d. The site included the buildings within the precincts, a mill, a flax field, fishyards and the ferry, ferryhouse and boat. (fn. 66) The priory buildings were allowed to fall into ruin after the dissolution, apart from the chapter house which was retained in use, first as a domestic chapel and later as a chapel for the extra-parochial district of Birkenhead until the new church of St. Mary was built on the site of the priory graveyard after 1819. (fn. 67) The ruins were purchased by public appeal in 1896 and their care entrusted to the corporation of Birkenhead; in 1913 a faculty was obtained to renovate the chapter house and it was dedicated for use as a chapel in 1919. (fn. 68)