From Hamilton Square you can take a train direct to Liverpool Lime St. No problem with bikes on Merseyrail, most trains have a section reserved for bikes & prams and no one will make a fuss unless you block an exit.
From memory there are big passenger lifts at Hamilton Square rather than escalators. At Lime St there are both.
Riding through the tunnel would not be allowed at that time & I can't see it being a pleasant experience ....
"Can I ride my bicycle through the tunnel?
Bicycles are not permitted through the Kingsway Tunnel. Through the Queensway Tunnel, they are not allowed between the hours of 6.00a.m. to 8.00p.m., Monday to Friday, 7.00a.m. to 8.00p.m. on Saturday and 8.00a.m. to 9.00p.m. on Sunday. Please note, the time limits during a Sunday are only operational between 1st April and 30th September inclusive."