For carving and general bushy tasks I find I would sooner grab a smaller hatchet instead of my SFA, but I have also used the SFA a lot for carving when I did not yet have the luxury of choice, and it worked fine. The handle gets in the way sometimes though. Your gouges are fine for hollowing the bowl, but I would strongly advice to clamp your spoon in a vice or workmate while using them to hollow out the bowl. The scar on my thumb can tell you a nice story of the time I thought I didn`t have to do that.
I do not think you`ll have to invest a lot in a straight carving knife at first. I find that the mora 511 with the guard cut off performs brillianty, especially if you are just trying to see if this is something you want to get into. The mora crook-knives are good as well, but require some work on the edge when you get them. If you can indeed get your hands on a Ben Orford for only a fiver more I`d be sorely tempted...
I`d say have a go at the birch with the tools you have! You`ll notice soon enough what tools you need in that process