I did initially put it together this way, and then decided on something better. When I placed my initial order with Adafruit, they sent me a free "perma-proto" board (https://www.adafruit.com/products/1609). This is nice in a couple of ways. First, it's about the same size as the Arduino Uno board, something I'll come back to in a moment. Secondly, it has very fine tracks and plated through holes, which makes soldering to it easier than stripboard. If there is a piece of advice that I would offer my teenage self just getting started in electronics, it is this: relish your youth. When you get to the your 50s, it's going to be very hard to focus on small things, and soldering will involve much guesswork. Thank you, my teenage self will say, now can you tell me how to get more sex?
Because the perma-proto board is about the same size as the Arduino Uno board, you can easily strap them together. I did this with a piece of cardboard between them for insulation, holding the whole thing together with a large rubber band I find in my desk drawer. I had a couple of spare headers which I soldered to the perma-proto board and then connected the 5V, Ground and input pins from the Uno board with ordinary jumper cables. The battery clip goes to the Vin termainal on the board, again connected through a spare bit of header. The result, while not pretty, is just about robust enough to be carried round in a pocket. It'll do until I've ordered a more rigid box instead. Here's a couple of pictures.