Eric, spending a night not only allows acclimatization, but also puts you in perfect position to assess conditions at the base while you're (presumably) fresh & ready to go. By conditions, I mean both overall with respect to weather, and of course specific to snow/ice.
The Chute is to the west (right looking from Trail camp) of the 99 switchbacks. You want to be at the base at first [strong/warm] light - say between 6-7am, so that the snow/ice begins to just barely soften a bit. If it's still icy, then it will be hard to kick-step with your crampons, plus it's dangerous as hell if you slip ie tough to self-arrest if your axe can't bite, plus nice & slick for maximum bodily velocity. So, hang out for 5-10-15 minutes while constantly checking conditions. Once it's soft(er), you can both kick-step + bury your ice axe as an anchor on each stride.
Overall, you need to get up to Trail crest, hike the 4 miles (round-trip) along the snow/ice western ridge to the summit, and get back to the Chute before it begins to ice up again in the afternoon.
For Newclimber, the main trail as an early season day hike with snow/ice conditions is pretty tough. In fact, I wouldn't recommend it. (Experienced climbers) can do the MR as a day hike since it's only 5 miles to Iceberg. In other words, you can leave the parking lot around 3-4am and be at the base of the Gully between 6-7am - just as snow/ice conditions get perfect. Likewise on the descent, since the Gully faces due east and has the Notch above it, the sun stays on the snow until the afternoon. So, it stays soft enough to safely plunge step back down. (Of course, this assumes it's a nice sunny day - if a storm or cold front is moving in, then conditions can obviously change for the [much] worse.)
When we did the MR last year, we spent one night @ Boyscout, and I was thoroughly worked. But, since the main trail is approx twice as long as the MR, and the conditions on the Chute are more difficult than the Gully in the afternoon (it faces north-east with Mt Muir casting a large shadow), you're going to be way more tired - maybe exhausted - just when it's most important to get back down that sucker to safety. Perhaps the biggest problem is you won't have an out - you can't use the switchbacks, and you can't stay on the ridge. So you have to get down.
You sure you want to do that, rather than spend the prior night within 1 mile of the base of the Chute, have a nice relaxed start, and then get back down to Trail camp by 1pm? If the issue is an overnight permit, I can practically guarantee you can get a walk-in before June.
Here's the Chute from the Lone Pine side of Trail crest from a few years ago around June 5. It's too melted out to descend (safely), switchbacks just opened (still lots of snow/ice, especially at the Cables, but passable):
