I think the idea is a good approach.
PVP:
Although I seldom play PVP myself.
To raise the value of PVP games, you could get a daily payout from BFP. The higher you are in the rating, the more you get. Maybe for 1st place then 100 BFP and then down for every 10 places 2 BFP less. Although I think it's important that you at least always get a certain amount, even at number 1000. So that even low elo players receive something and have an incentive to play up.
I had read somewhere here that the prices, as well as rewards should be raised, so you get a little more diversity. This would be even better for PVP rewards, as you will receive 1.000 BFP for 1st place and then less and less.
(r/c) PVE
For PVE, you could set a daily limit on how much BFP you can get in PVE games.
As a prerequisite, you would have to play a number of minutes and no number of games, so no games arise that only go for 3 minutes.
1 BFP per minute, maximum 50, so that PVP will be rewarded a little more, because theoretically PVE can simply be farmed off and you can already get the gold here.
But I find it difficult here, as this already overlaps with the daily quests.
OR:
Or just take out the daily quests and take the PVP and PVE rewards as described above. So both types of players would be picked up for both types of play.
My personal opinion is that more players would come to the game through the PVP, as at the present time, simply the PVP is more prevalent overall and much more is shown on public media like Youtube or Twitch.