Which one would you choose Cliff, or which one do you feel is the better knife?
This is an interesting choice because the knives are fairly different. The biggest point of contention is that the F1 will have more steel around the edge. Both of them run full convex grinds but Fallkniven runs them more convex so there is a heavier edge curvature and they also apply a secondary edge bevel. Personally I would just add an edge relief to the F1 specific to my demands, however if you are not going to do this then for most wood craft you will see a benefit to the thinner edges on the Bark River.
However there are some factors on the F1 which compensate. Note the large choil on the Gameskeeper, that isn't something that you want for wood working in general, outside of really precise work. If you work with your finger in the choil you are basically working on a horrible handle shape because it is a piece of flat bar stock and at best is radiused, the guard as well looks to present a less than ideal resting point for the finger. It is a high pressure spot, has little security, control, etc. .
This would cancel out the inherent cutting ability advantage for significant carving (roughing wood to shape, pointing and similar). So I would look at handle considerations to decide and the F1 takes it there easily as the grip is more secure and the extended tang provides a point to hammer with and on and this is useful in a lot of wood working. Fallkniven themselves, through Peter, does a lot of such pommel impacts for various tasks such as splitting wood, felling wood, cutting to length, etc. .
The stock thickness of the Gameskeeper is also more than a little absurd considering the length of the knife. You really only need that thickness in a small parang which is designed for thick wood chopping and most of them are even a bit slighter as they are usually 0.2" or so. The spine thickness isn't a huge drawback to a lot of wood work though as that tends to be more focused with the cross section at the edge directly, but it would limit the performance of the knife pretty severely for general utility. The only real reason for this would be if you do a lot of thumb/finger on spine cutting and need the extra width for ergonomics.
In terms of steels there is a large difference as VG-10 is a high carbide stainless and A2 a relatively low carbide tool steel. Expect a greater ease of sharpening A2 to a high polish but if you run mainly coarse edges then it won't have a significant advantage mainly as it is significantly underhardened in the Bark River knives.
In short, I'd buy the F1 for those tasks you describe but these knives are so different and there is some give and take in performance that it would not be unexpected to have opinions swap. In contrast if you asked about something like the RD6 vs F1 you would expect a much more lopsided responce as those knives are built to almost completely different goals.
Whichever one you go with I'd be interested in an email with your persepctive after you give it some use.
-Cliff