The third option is Excess/Checked baggageBrainless as my link explained checked baggage is not possible.
It is a VERY long journey to drive via channel tunnel. When we left, we stayed in a hotel a couple of miles off the tunnel so we could set off first thing the following morning. (We live 4 hours drive from Folkestone.) It took us 15 hours, with just 2 very short stops, to get to Southern Sweden. We then stayed overnight with a friend and the next day drove the next 8 hours up to Stockholm. On the way back, we wanted to go all the way home without using a hotel, so started off in Southern Sweden again, left at 5 am, but hit serious roadworks (and 34 C heat) in Belgium and France, missed the tunnel train we had booked, and the entire journey ended up taking us 27 hours -not counting the trip from Stockholm and down south. With a pup it would necessitate a lot of stops and the important thing is then to check with ALL the other countries involved (i.e. France, Belgium, Holland, Germany and Denmark) if
they allow pups NOT vaccinated against rabies to be out. With the kitten it wasn't so bad as she stayed in a big cage all the time and didn't need to leave the car. The Swedish ministry did say though the person who imports must check with all countries they drive through.
If I'd had more money available at the time, I'd have taken the ferry to Denmark from England.