Alex, I'm surpised you prefer to use your tent without the fly. The 'rainfly' is for more than just rain. It also keeps you warmer by minimizing wind and it gives you percieved feeling of safety from animals. Once in a while I use my mesh tent without the fly, but only when it's warm, calm, 0% chance of rain and I want to see the stars.
If you normally used the fly, you wouldn't have to worry about getting doused with rain in the night.
Regarding the fly getting wet, condensation occurs in any tent in the right conditions. You can minimize this with good tent design and setup, but it's not totally avoidable. The best is to get in the habit of folding your fly up so you keep the moisture inside while hiking instead of exposing the rest of the tent to the moisture. If you do this, the tent body should be dry the next night and if you set it up early the inside of the fly can dry too.
I'm a little confused at what you are looking for. A fabric bodied tent would likely give your dog peice of mind but you'd still be at risk of getting rained on, and you'd get condensation inside the fabric if you didn't use the fly. So really, you'd normally want to use the fly anyways. Perhaps you just want another mesh bodied tent that is more spacious, durable and has a few windows? You can find some nice 4-5lbs 2 person tents that offer this. If you don't mind less waterproofing and no windows, the Scarp 2 is quite a light offering.
Regardless of you choose, I would plan on using the fly most of the time for condensation reasons and to avoid a mid-night rain soaking.
You could buy single wall tent, but then you've got an increased risk of getting wet because the condensation will form inside this instead of inside the fly.