You can slip a stitch onto the other needle to sit alongside the beginning of the round stitch and then just start working with your smaller needle. The first few stitches can be a little tricky to hold everything together but you can also put a stitch stopper or point protector on the larger needle to stop the stitches dropping off the unused end.
Another option is to work the first row flat (working off the larger needle and onto the smaller needle) and join on the second round once all the stitches are on the small needle. The little gap of one row can be filled with sewing in the yarn tail when finishing up.
Another way is to keep the needle you want to work with (the smaller) but add a second needle alongside it, holding the two together as though they are one. This can be a smaller needle than the one you want, it is there to make each stitch karger without “trying” to change tension. Once the stitches are cast on you pull out the second needle leaving all in the working needle. This works well for long tail cast on. I don’t use long-tail cast on so I don’t use this method but use seen it.
There are also other ways to cast on, some stretchy, some very attractive, perhaps something to look into at a later date as it’s probably not what you want for this project.