A change in the URL may or may not go unnoticed, but different designs, navigation menus, layouts, logos, etc would be disruptions.
As an example – there are usually huge abandonment rates when sending people to a PayPal hosted checkout/payment page. It looks different, it’s not the design/style/font size/etc. they were used to and it breaks their mojo, causing confusion or introducing the opportunity for second guessing and buyer’s remorse before the purchase.
BUT, integrate with PayPal’s api, roll your own payment solution on your own site using your layout, fonts, etc so it is seamlessly integrated, and people will be more likely not to abandon.
It’s not that you’re more trusted or known than PayPal, it’s the interruption that snaps them briefly out of buying mode. That tiny “what just happened” moment introduces just enough doubt/hesitation/concern in people to cause more of them to abandon the purchase.
It’s UX and buyer psychology, and why I suggest that it comes down strictly to dollars and cents. If you’re going to sell $50/year of merchandise, it’s not worth the $101. UNLESS you know that you’re losing more than the $101 each year in sales.
Analytics will help you here. Do a good job of tracking what people are putting in their cart, how they’re navigating the site, and if/when they’re abandoning. If it’s happening during your transition to the other site for checkout (and happening enough to justify the $101/yr), you’ve got your answer.