Woodside ferry terminal (Birkenhead) is currently closed for refurbishment, so you would have to ride to Seacombe, about 1mile. You said you were arriving at 6.30am so you should be okay, but be aware that the commuter ferries only run from 7.20 to 9.40 in the morning Seacombe to Liverpool. The evening ferries from Liverpool to Seacombe run from 17.00 to 18.40. Tickets cost £2.90 one way. The only other tickets are the river explorers at £11.50 a pop, one way. Don't be like me and turn up at Seacombe ferry terminal one sunny summers day with about £7 in loose change heading for a cycle tour of Liverpool!
If you did fancy going the long way round be aware there is a dedicated cycle lane on the Silver Jubilee bridge at Runcorn though it would be a bit of a faff getting there from the far side of Ellesmere Port.