Thought I'd report back on my experiences on this thread as the OP, in case of interest to anyone searching in future. I was indeed able to book tickets in the end, about six weeks before I travelled, though I decided that an extra £5 each way for an open ticket as opposed to an advance would be a worthwhile investment.
I travelled out on the 20th using EMR to Crewe, TfW to Chester and another TfW train to Holyhead, then the early afternoon Irish Ferries sailing. On the way back I took the Stena Line morning ferry, then an Avanti train from Holyhead to Crewe and EMR back to Tutbury. In both directions the journey went without a hitch. I decided I would take my folding bike. On the way out, the person at the checkin desk for Irish Ferries said that I needed to check it in. I didn't argue, but I was slightly trepidatious. Fortunately it came out the other end unharmed. I think I could have 'smuggled' it on myself without challenge given the way things are set up, but I decided not to. On the way back, Stena were OK with me taking the bike as hand luggage, though the person at the desk told me I would need to keep it close at all times as they were expecting the boat to be busy. It was indeed quite busy, but the bike doesn't take up much space and it was no problem keeping it close.
Cycling from the port into town was fine. It is reasonably well signposted and there are even some cycle lanes, though these are mostly shared pavements with pedestrians. Arriving in Dublin at about 18:00, I'm certain I was much quicker to the south side of the city centre than a bus or a taxi would have been.
In terms of the ferries, Seat61 suggests that the Irish Ferries vessel is the better of the two. For me there was very little in it - they're both pretty large boats, reasonably well equipped, and with space for a stroll as well as good amounts of seating. If anything I had a slight preference for the Stena ferry - it didn't feel any smaller and the wifi was free (only paid options on Irish Ferries as far as I could see). The food also seemed slightly cheaper. There were quite strong winds on the way out, but the Irish Ferries boat felt very steady; I'm not sure whether the Stena boat would have been as secure.
Trains-wise, I was surprised by how busy the mid-morning shuttle between Crewe and Chester was (full and standing!), and the train from Holyhead on the way back also filled up very quickly before a second unit was added at Chester.
Overall it was a good experience. That said, there's no denying that it's a much longer journey than going by plane. It's a day's travel each way, and it was quite tiring. If I was making the trip regularly I'd be tempted by Birmingham or Manchester airports, but for the occasional trip, I'd definitely do it again. It's great value and an interesting journey.
