I lived between Santa Fe and Bandelier for a few years and know the area well.
The trouble with the Bandelier area is that Los Alamos National Laboratory is right there. It's a jurisdictional nightmare and therefore there aren't many campgrounds. There is one west of the Bandelier entrance maybe 25 minutes away called Jemez Falls Campground. Here is the link:
https://www.fs.usda.gov/recarea/santafe/recarea/?recid=75776. There there is one a bit further than that called Redondo Campground but that's getting even further out maybe 35 minutes west. Here is the link:
https://www.fs.usda.gov/recarea/santafe/recarea/?recid=75806. Because I lived right there, I've never stayed at either campground because I usually 'boondock it', and I never ventured over there on July 4th weekend. I am sure it will be fine, however.
I think you'll have more trouble finding camping after visiting Pecos NHP. The NHP is just south of the town of Pecos, and the town is just south of the Pecos Wilderness. NM-63 is the road that runs north/south connecting them all, and which dead ends at the wilderness. There are lots of campgrounds between the NHP and the Pecos Wilderness - public, private, some facilities for big trailers, etc. - but they are really, really busy. This is where all the extended families from the area go to have big gatherings and fish and so on. I don't know what to say to expect, again, since I lived there, I would never go to Pecos during that time lol. However, here is the forest service website which lists the campgrounds:
https://www.fs.usda.gov/activity/santafe/recreation/camping-cabins/?recid=75436&actid=29. The relevant district is the Pecos - Las Vegas Ranger District. I don't think any of the public campgrounds are reservable. If you wanted, it would be a great place to rent a cabin for one or two nights, Pecos is a gorgeous area. Lastly, I am not sure how to find private campgrounds online.
In conclusion, I think you will be fine at Bandelier. At Pecos, I believe it will be tougher to find camping. The distance between Bandelier and Pecos NHP is about 1.5 hours, so it may be best to get a site at Bandelier for a few nights and day trip to Pecos NHP.
Hope that helps.