The Biolite will work. goTenna has used them before for some experimental installs.
I’ve been using a certain battery pack I get from banggood.com and it has proven reliable. It has the instant on feature which supplies power when it senses a load. In any case, what you want is a battery pack that can both take a charge and supply a load at the same time.
Recently, I found a new battery pack and found it to be every bit as good as the one I’ve just mentioned and written up several times, and it’s priced a little better, too, although both are very affordable. You can see it here:
For the panel itself, I recommend going with at least a 6 W panel if you’re around 40 degrees north latitude.
Yes, putting your relay outside the attic space will improve the performance, as even the roof being wood and shingles it will still attenuate the signal some. Plus getting outside on the roof will raise the device just a little even if you don’t raise it on a mast. You can see how I did my home base install in the Urbana thread as an example. However, modding the small water-tight boxes can provide a lot of flexibility to meeting your needs however you go about this.