Great ideas and questions already.
If you have to go see Titcomb but only have 4-5 days your choices are set by your pace. Green R or New Fork to Island L to Elkhart plus day hike to Titcomb is close to 50 miles. Do you want to do 10 miles/day with about +9229/-10520′ total gain/loss?
The easiest is an out and back just to Island L. 24 m round trip. Add in another ~5-7 m for Titcomb. So, 12m 2600' gain, 1600' loss just to get to Island Lake from Elkhart. Either a long day (with full backpack) or 2 day entry. Walking out is easier since packs are lighter, but it's still a lot of up and down.
The alternatives
@Bob gave are spectacular. Still long approaches. Could basecamp near the Cirque if you don't mind crowds. Easier walk in/out to basecamp anyway. But doing Jackass Pass twice in one day wouldn't be my choice. Very nice area near Deep Lake though. I've done that when I only had 3 days to spare. Or, camp at Big Sandy night 1, spend 2 nights in the Cirque and a long day on day 4 back to the trailhead.