What you've said is correct, that the feature to integrate the PayPal app is only available for QuickBooks Online (QBO).
QuickBooks Self-Employed (QBSE) is designed to help self-employed individuals in tracking their business transactions. While QBO is intended for small and medium business owners. For a quick view of feature comparisons between these 2 products, you can view this page: QuickBooks UK. Scroll down the page until you see the product comparison table.
Since we're also talking about 2 different products, the option to migrate from QBSE to QBO or vice versa is unavailable. If you plan to use QBO, that means you'll be starting a new company from scratch.
If you decide to continue with QBSE, then I'd suggest using the other alternative you mentioned. And that's to send the invoice straight from PayPal Business. You can record the invoice in QBSE as well, so you can keep your records updated.
Should you wish to decide and use the QBSE mobile app, you can view this article: Overview of the mobile app. It provides a short introduction of the app, and the features available in it.
Please let me know if I can be of further assistance by leaving a comment below. I'll get back to you as soon as I can.