Comparing results of a promotion camapign: Control vs Treatment Sub-Group

#1
I'm looking for some clarification on the correct way to test the difference between groups.
At my company we ran a promotional campaign for credit card customers with a chance to win a prize. We took all customers who had a credit card and divided them into a Control Group and Treatment Group.
The Treatment Group would recived an email with campaign details and a link to register for the promotion, the Control Group would not recieve anything.
To be eligible for the prize, a customer would have to register for the promotion. The promotion was aimed to increase customer's spending during the promotion period.
Now I want to check whether the promotion had any impact on the customers' spending behaviour.
I have 3 groups,
1. Control Group
2. Treatment Group who registered
3. Treatment Group who did not register
I would assume that the first test I need to perform is a two-sample t-test of Control vs Treatment.
But does it make sense to run a two-sample t-test on Registered customers vs Control? The way I see it, the Control group had no incentive to change their spending behaviour but neither did the non-registered customers from the Treatment Group. I want to say that those who know about the promotion, do spend more as a result of the promotion. Does this type of comparison make sense?
Thank you