Went to Bryce/Zion at the end of our 2006 Yellowstone trip, when the kids were 7, 11, 12, in mid July
We arrived at Bryce around 7pm, stayed for the night, and did a brief tour in the morning, out to the point and left. Had not planned on more than that at Bryce because the park is similar to Grand Canyon, great views, but not much more unless you are going to hike into the Canyon and I knew the kids would not be up for that kind of hike late in trip. Did not feel I missed out.
We had two nights booked at Zion, with the intent on a day hike up the Narrows. We brought water shoes especially for the hike, but in the future, would just rent them. Best hike of the 4 week trip. My 7 year old liked it so much he wanted to do the whole 16 mile Narrows hike from top to bottom, which we did in 2011.
Given you won't have more than a couple days, that's what I would do, hike the Narrows, with maybe a quick look at Bryce.