Found this on What Pub.

Probably the same as the Hindon Hotel and Hindoo Tavern at this address in 1850 to 1860. The Hindoo Tavern is listed in Bagshaw's Directory of 1850.
(Listed in the wiki index as '28 Hindon Tavern Beckwith Street')

The Pub was leased to Anchor Brewery from 1861 and transferred to Birkenhead Brewery in 1866 as the Bank of Ireland Hotel, it possibly got its name because Irish sailors picked up their pay nearby and then spent it in the Pub.

The photos from the 1950s are courtesy of Helen Wood whose grandparents Frank and Mabel Ryan were licensees from 1948 until the time the Pub closed on 22 June 1962. The interior shot shows Frank Ryan behind the bar on the far right. The Pub was later demolished.

https://whatpub.com/pubs/WIR/479/bank-of-ireland-hotel-birkenhead

Images.
https://whatpub.com/img/WIR/479/bank-of-ireland-hotel-birkenhead/298/224/22025883
https://whatpub.com/img/WIR/479/bank-of-ireland-hotel-birkenhead/298/224/22025884
https://whatpub.com/img/WIR/479/bank-of-ireland-hotel-birkenhead/298/224/22025885