This could be a long post,
Assuming here Riot are adamant Franchising stays.
Increase the Masters events to 3 (12 teams) and Champs (16 teams) will remain the end of season tournament, say the season starts 1st February,
Have 2 or 3 week regional qualifier (like kick off, not a league) where all Franchised teams take part, along with 1 or 2 tier 2 teams will have qualified for this qualifier. Top 3 quality for Masters.
Masters starts end Feb / early March, 2 week tournament, no need for it to be longer and no need for a big gap between regional qualifiers and the start of Masters.
2 weeks after the final of Masters 1, say late March, the same process starts for Masters 2 (tier 2 teams should have a way in), this gets to in or around late April / early May, once Masters 2 complete's, there is a break until say 1st June.
1st June, same process to qualify for Masters 3, (tier 2 teams should still have a way in).
Once Masters 3 is complete, the regional championship starts, not a full league but more matches than the qualifiers for Masters 1,2 & 3. A couple of tier 2 teams should be able to also qualify for this. The winner becomes the regional champion, qualifies for Champs, along with the runner up.
The remaining 2 spots for Champs can be done on a points system, not perfect but it does reward some form of consistency through the year, but only have points awarded for qualifying masters 1,2 & 3 not for actual wins losses in the regional qualifiers.
This all effectively means Franchised teams are automatically in the qualifiers for the Masters events but that tier 2 teams have a chance at getting into those qualifiers, so if an Acend or a TSM make a push with a good roster, the have a chance at regular reward.