Barge cement is a contact cement. Just a different brand. Weldwood would work fine if that's what you have.
Depending on what I'm working with, I don't always hold the two surfaces apart. For example, given what you want to do, it would be a good idea to mask the riser near the grip so you don't get cement on it where you don't want to.
In this case, I'd just get some contact cement between the two surfaces as quickly as possible with a toothpick, for example, then immediately press them together to spread the glue around, but let the leather relax/pull away from the riser for 5-10 minutes and then press it in place. No problem if some of it sticks right away. Just be sure to press everything together about 10-15 minutes after you apply the contact cement. It's really quite easy. Good luck.