What would you think of a league where you bowled half of it handicap and half scratch?

Say 1st quarter handicap, 2nd quarter scratch, 3rd quarter handicap and 4th scratch.

or

Have a Handicap league that has more position rounds, but you bowl the position round scratch?

If it's an existing scratch league that you're trying to integrate handicap into, you'll likely lose teams. Scratch bowlers generally hate the concept of handicap, and many will go elsewhere if there is a choice. I fall into that camp.
 
On the other hand, handicap bowlers generally are fearful of the word "scratch", and won't vote in rules that allow it in any form.
 
I've never heard of that kind of format, and would be surprised if I heard any real life success stories.

It has debated here ad infinitum but in my mind average caps are still just a form of handicap. A way to make all teams fairly equal.

This league was not capped, was dominated by 2 or 3 top teams, and was top heavy on the league payout.

But at least around here most of the 200 to 225 crowd, the bulk of the"scratch" bowler pool, would rather beat up handicap bowlers than bowl scratch against the 235+ bowlers.

I'm skeptical that you'd get a lot of support for that concept because, as others have said, most bowlers plant their flags firmly on one side or the other (scratch or handicap). However, I guess it's always worth giving it a try if you can find people who are willing to do so. If I were you, I would just make sure that you have every possible rule and detail locked down before proposing it.

Personally, the best format I've come across (other than straight scratch  ) was one that we used in Texas. Your bowling order was determined by average. Your top average bowler had to bowl 5th, your next highest bowled 4th, 3rd-highest bowled first, 4th-highest bowled 2nd, and lowest bowled 3rd. Then, you bowled both team and head-to-head game to game and for series. Meaning, not only were was your team trying to win the three games and series, but you were trying to win individual games and series from your opponent. Within the overall standings, your individual winning % was listed, so there was that added pride factor in maintaining a decent winning percentage. I liked this format a lot because the individual accountability game in and game out encouraged bowlers to keep trying to win their individual stuff even if the team was already way down or way ahead. All in all, I felt like there was less bagging in that scenario than I've seen in most others.

In any event, good luck with your scratch-handicap hybrid league. I'm all for seeing new things tried out in leagues. Let us know how it goes if you can get it started!