Have been a big fan of this for years, back when it fell on deaf ears.
Personally, I am of the opinion that prizes should be directly proportional to the number of matches played, 16 players = 4 matches = 1 Mark code for first place as a starting point. If you play less matches because of a bye, your win chance increases accordingly, so I think it is not unfair to reduce the amount of points gained due to a bye. You still always get a trophy for first, so it is always better to have a bye for a better chance at rewards.
For second place, about half the points seems right. Third is barely worse than second (same amount of matches won), fourth is just bad and should not get any reward, or barely any reward. Great contrast between third and fourth helps motivating for playing third place match, too.
If it could be merged with pvp participation stuff like suggested, that's even better in some regards. Keeping it equal in terms of points gained would decimate the tournament rewards, however, unless you keep the first place reward. Keeping it means your entire reason for doing this is gone and you should not really allow single-elim 7- player tournaments. Additionally, pvp events include lots of matches, tournaments just a few.