Is Soleil worth Distant Counter?

Other than arena points, I don't think so. She shines as a player phase unit, and there are much better candidates for DC.

She can use it pretty well, she has a pretty similar statline to Ryoma, and since you're going to +10 her it would be a decent investment, you might want to patch up her def a little though since you have a -def IV

If you gonna run dc, she's better off with vantage.

The thing about Sunshine is that she can excel at any role with her statline and the right IVs. If she's going to be your +10 project (yay for a fellow Soleil fan!), then DC can be a good investment. Ideally you'd not want her Def as the bane since it's a superbane, but if you can get a -HP one you're in good shape.

If you do go for DC, consider wrath and Aether as the B and Special. Helps her heal a bit more frequently and dole out more damge. DC + Wrath has been a great combo on my Lucina, and it should work just as well on Soleil. Vantage as Drizzy suggests also is potent.

She'll be your +10. If you'd like her to be a little more of a swiss-army knife, go ahead! She can carry it just fine. Just might want to aim for a better bane. Hope that helps!