Try to get those last minute permits right at 5 p.m. the night before if your day already has permits sold out, but have a plan B (like it sounds like you are developing if that doesn't work out for you - very smart). I like you, heard that they sell out in minutes and heard that some people practice using the site in advance to try to get these efficiently too.
Other than Bear Lake Corridor (and you are right that this has a lot of the great hiking) and that is the area where you have to enter before 5 a.m. or after 6 p.m., you can access other areas of the park before 9 a.m. and after 3 p.m. without a timed entry permit. Just wanted to make sure you understood that.
https://www.nps.gov/romo/planyourvisit/timed-entry-permit-system.htm
And the other areas of the park include that great drive up Trail Ridge Road with all the stops and overlooks along the way up to the Alpine Visitor's Center which is the number one thing to do in the park (It's that much fun and that enjoyable). I would plan on an early day to do that if you are unable to get the timed permit. I'm not one that likes long drives or likes to do long drives with kids, but with all the stops and overlooks this does not feel long at all and is a fabulous / fun day for all ages (recommended for a day other than your first day to acclimate a little to the altitude since you are going over 10,000 feet).
Another area that is highly recommended for kids (family friendly hking and playing around) that is outside of the Bear Lake corridor is Alluvial Fan.
https://www.tripadvisor.com/Attract...an-Rocky_Mountain_National_Park_Colorado.html
The wonderful Estes Park Visitor's Center is worth a stop for ideas of fun things to do with your kiddos. It's in town and you don't need park access to stop there.
https://www.visitestespark.com/plan/visitor-centers/
I'm not sure with your young children if this would be of interest or not or work, but we just love stopping at the Stanley Hotel for the views. We usually like to do the architectural tour there. This would have been fine for ours at age 6 (age 2??) You could take a look and see what you think.
https://www.stanleyhotel.com/historic-hotel-tour.html
The only place that I visit more than Colorado is Orlando lol.
I love Estes Park / Rocky Mountain National park and we will be going again this year in end of September mid week (for slightly less crazy crowds -- late September is elk rut season and brings out lots of people). I don't live in Colorado now, but used to live in Denver. I live in St. Louis now, but do vacation in Colorado pretty frequently.