It would be easy to continue in several ways:
A. Same as before. Minimum contracts don't count against the cap until after the auction, you have to outbid if you can, must outbid by 100K, if there are matching bids because that is all that can be offered, mini-roto decides. If you have the money the difference between 250K and 350K isn't very large, and if you are close to running out you will be in the same position as those who spent their cap.
B. Teams with money left are allowed to match offers from teams who don't (without having to outbid) and the mini-roto decides.
C. My preferred option: No exceptions to the cap until after the auction. In this case, minimum contracts should count against you. Teams over the cap must wait until the auction ends to fill their roster using the minimum salary exception. Auction ends when every team with cap space has filled their roster. Those who spent their money likely planned on being able to fill their team with minimum contracts during the auction but this gives you an absurd advantage.
All work and are close enough to being fair in my opinion. Nothing is going to be perfect.