Is there a way to find a simple regular expression for this language?
Forget about the $0$, they are boring. Like you said, you can add $0^*$ basically everywhere and you are done.

Now for the $1,2$. Just try to order them in batches that total 5. Then you realised that this might be problematic in case at the count of 4 we read $2$ and overshoot by one. No problem, we remember this, and continue until we reach a proper multiple of five. Note however, that the overshoot just might repeat when we have $212$, or similar.

So here is how the regular expression is constructed: any multiple of five, or when we reach an overshoot, wait until we are back. Repeat.

First some abbreviations:

$E_5= \{11111,1112,1121,1211,2111,221,212,122\}$.   

$E_4 = \{1111,112,121,211,22 \}$.

$E_{\mathrm{shoot}}= \{1112,212,122\}$.   

Finally here is your expression:

$\Big ( E_5 \cup E_4\cdot 2 \cdot (E_{\mathrm{shoot}})^*\cdot E_4\Big )^* $.

Ah. Add $0^*$, lots of them.