If he had his own bike Julie could use that to cycle to & leave at the pub, prior to getting the train home. So he didn't need June's bike.
Back in the 80's bikes tended to be for women or men. I didn't like riding mens' bikes as they tended to have a bar from the seat to the handles that tended towards FGM if you straddled it! Especially racing type bikes. I think they were deemed a bit unladylike too as you had to swing your leg over the back to get off. Whereas with a ladies bike the bar was set at an angle and there was no need for leg swinging as you could bring it across the front.
Whether Bamber had one or two bikes, he would still have to either drive or walk to the pub to collect the bike. He couldn't use the second bike to collect the bike at the pub as he can't cycle two bikes simultaneously.
If he wanted a bike to get around on before he had a car he would simply ride to a bus stop, train station, pub, shop, party whatever and leave his back there ready for his return journey to WHF.
If he wanted a bike as his mode of transport to murder his family he could simply walk to work one morning and ride it back to his cottage.
Why bother with June's sit up n beg type bike when he probably had his own bike?