No, that setup would not work if you are resident in EU.
Strictly speaking, you can only avoid VAT on sales to non-VAT registered EU persons by not selling to them at all. Even non-EU companies are required to pay VAT.
In reality, a lot of non-EU companies are selling digital services into the EU without charging and paying VAT. This works fine as long as you don't have any assets within the EU that a local tax authority can lay claim to. But if your US LLC is managed and controlled from within the EU, it's an EU company as far as tax is concerned and you need to pay local tax (corporate tax, and VAT).
Your structure looks like
Tax Evasion 101 and won't fly. If your business model can't handle VAT, I think the problem is the business model, not the VAT.