It's like this: if you don't have the chance to check out the tattoo place beforehand, you're not going to want something elaborate. If you want something elaborate, you're going to want to be able to check out the shop beforehand and have the ability to possibly go in for multiple sessions.
It sounds like the latter is not an option for you, so you will want to keep it very simple. I would recommend getting the date tattooed while you're in Cardiff, and then have the red dragon done at a shop that you can check out and return to if needed. You do not want a rushed tattoo, especially a memorial tattoo. I don't think that caring for the tattoo afterward is going to be most of the problem - just make sure the ink can breathe and you keep it clean. You treat it like you would treat a sunburn, for the most part. You make sure no clothing rubs it, you make sure not to scratch it, and you keep it clean and moisturized.
However, the difficult part for you is going to be getting quality, safe work with the deadlines you've placed on yourself. It's my understanding that red is tricky, and can sometimes take several coloring sessions to come out nice and vivid. It's also my understanding that a tattoo like that is going to take more than one session to complete. I think you're setting yourself up for disappointment if you try to go for that. It might work - it might be fine. You might be able to do it all in one go and have it be fine, but that will be if you are lucky. Since this is a permanent mark on your body, you want to give yourself the best chance of it looking fantastic for the rest of your life, which means planning for the time that you have and sticking to it.
Getting simply the date in black ink will be really hard for the artist to screw up, and it will be much easier to care for while you finish your bike trek. If it were me, having already gotten a tattoo and having seen what goes into even my simple shaded tattoo, that's what I would do.