It's completely legal. Think of the following situation instead (since the principle is the same):
Your company A does say...project management. You have a contract with another company B where it's stated that all expenses are to be paid by company B. So company A hires say...architects, construction workers etc. and then later invoices those expenses to company B in addition the the fee for it's services. Isn't company A technically a payment processor for company B in that case as well? Hint: It practically is, this is perfectly legal and requires no license.
Again, this is just an example of a completely different setup in a different scenario, but I'm just trying to show you another situation which is a lot more common and involves a company technically providing payment processing services.
In your case, you only need the SVG company IMO. Keep in mind that if you go the "payment processing" route, the payment processing company should probably not be a subsidiary.