It looks like you're following the Wirral circular trail, which is a mixture of shared use pavement next to the road, some nice bits through Eastham and some quiet roads and industrial roads. I've never done it but as you say, it should get the job done.
All the trains have cycle spaces at either end of the train, marked by a bicycle on the door. So the three train carriages have spaces front and back, so it is definitely an option. Bache will probably be the closest one from the zoo.
I've ridden all the roads
@Sixmile and
@Crackle have mentioned, as I live near to Birkenhead and work at the Countess of Chester Hospital, right by Bache Station. Get the train from Bache and forget trying to ride it, as your only route choices are relatively poor, especially if you have a trailer with children.
There is only one good route that you can take, but it's quite long, around 35miles to get you from Chester to Birkenhead, basically you head back into Chester from the Zoo and pick up the Greenway from there and head towards Burton Marshes. There is a short stretch through an industrial park, but it's all cycleway and signposted. Once on the marshes you stay on them until Neston, then pick up the Wirral Circular and follow it to Seacombe, all around the peninsular. It is a lovely route, and totally safe, being 90% cycle / dual use footpath and one I often ride for pleasure, but being realistic, it will take you 3 hours or so with a trailer, possibly closer to 4, so with your need to catch a ferry, I'd take the safer option.
The route
@Crackle suggested is a good one, although that stretch of the A41 is very very busy around 5pm, and the cycle provision along that stretch is poor. There is a footpath you can cycle along, but its surface is shockingly bad, and the gutter of the road is often strewn with debris and glass. I ride primary all the way down that section.
Your original route is quieter, but again not one that I would choose to take with a trailer. The roads around the back of the Zoo are appalling, it is very rough and pot holed, and often dirty as well, so if the weather is wet you will have a miserable few miles. You should be fine then until you get to Ellesmere port and the route you've chosen looks like it goes down North Road past the Vauxhall plant. You have to pass under the M53 if that is the case, and the subway to do that, is quite tight and twisty, but manageable, however once you are on North Road, there are a lot of lorries and again the road surface is quite poor and littered with glass and other things which have fallen from the lorries, and remains like that more or less until you get to Eastham Country Park, at which point the ride becomes nice again, and you can stay on a good route more or less to Birkenhead.
I'm not trying to say it's a difficult route, because it's not, and for the most part it's rideable and I would have no problem taking someone on that route, but there is a raised risk of puncture, and with a trailer and a deadline I'd suggest the train.