As locals with concessionary passes and travelling with paying companions we've had alsorts/no trouble using Explorer cruise to 'just cross' the river, individual staff seem to think differently about what is allowed/chargeable! You should ring and ask what is correct and if you get a favourable reply get it confirmed in writing - try several times if needs be
PS Seacombe is the only terminal on the Wirral at the moment as Woodside (Birkenhead) is being upgraded (Seacombe having been done last year).
PPS Birkenhead Conway Park is best station to get off a train as it has huge lifts, James Street in Liverpool has large lifts only down to bridge level, lift to platform level is tiny, bike need up-ending so difficult with mudflaps. Wirral line from Moorfields Station in Liverpool can be accessed using escalators (let bike settle in position on a step by moving with bike, apply brake and relax - or fight the bike all the way down 😂).