msalley, I suggest:
1) Get a larger tank. 1G is better than half, but still too small. You want about 2.5G. Can you return the 1G?
2) Yes, most bettas don't like a strong current. Can you adjust the flow of the filter? If there's no adjustment, you could try blocking the output. I'm not sure what filter it is, but if you stuff the filter body with some filter material (floss, sponge, whatever) that'll slow down the flow.
If you can't get a larger tank and can't slow down the filter, you need to do daily 50% water changes.....