No clan chest after playing the team battle

The "lost cards on upgrade" is an illusion. (Well, poor interface design, to be percise.)
The upgrade animation shows draining of all cards, but the extra cards are actually saved. If you have enought cards, you can even upgrade a card multiple times in a row.

You should feel free to upgrade your cards without a worry. (Other than spending too much gold.)

Battling with no chest slots open doesn't have any real drawbacks. While you can't get a chest if your chest slots are filled, it won't move you in the drop-cycle. In fact you can actually use this to your advantage, as you can push to a higher arena or drop to a lower arena before a big chest to have a higher likelihood of getting certain cards. If you move to a higher arena you have the chance of unlocking the new cards from there, whereas dropping an arena can prevent you from getting certain cards from a chest. For example I got my giant chest in arena 5 instead of arena 6, because I didn't want to chance getting 100+ mortar cards that I will never use.

Also, you can still earn crowns to use to unlock your crown chest even if you have all your chests filled.

So if you want to get to a certain arena before a big chest, or if you want to to earn crowns, it's worth attacking with full slots. If you wish to stay where you are, and don't need to get crowns, the only benefit you will get is the fun of playing.