Yes, it is perfectly fine to switch tanks. My betta LOVED his upgrade from a one gallon to a five gallon. I moved over his old decor which he is familiar with and added some new decor. In an underdecorated tank, sometimes they can get a little freaked out with wide open space. I suggest that you add at least one plant in the new tank that reaches the surface or gets close so he has resting places near the top, and a tank that is wider than it is tall is best.
You will need to acclimate him to new, clean water. You should already be doing that when you do 100%s though so it will be roughly the same.
You don't need a filter. Many people recommend that you don't filter anything under a five gallon, anyways.
There are a lot of beginner plants that are hardy and don't require a lot of light. However, if you don't have dirt substrate, you need a nutrient supplement and correct lighting. I will find a link to a very helpful thread I posted a while back for anyone who is very inexperienced with live plants.
76-77 is kind of cool. They should have at least 78 degree water. A heater is pretty much always necessary for bettas. A 25 watt adjustable heater (like the Hydor Theo which is what I have) would be perfect for 2-7 gallon tanks.
Edit: Here is the link to that thread about plants:
If you have the patience to read it all it contains very good info on the needed light/nutrients and a lot of easy beginner plants as well as how to trim them and everything.