Hood River or Seattle/Leavenworth. You will have access to boat rentals in the seattle/leavenworth area, as well as Hood River (i believe the kayak shed has a rental fleet, pretty sure next adventure in portland does too). If you guys are doing the green "light", then you would be looking for class IV and IV+ rivers in washington. Keep in mind that most runs here are considered "in" with what you could compare to 200% on the green and "out" at a comparable 100% on the green. Just a heads up when you are meeting up with folks and asking about flows. Additionally, if someone says something is "too low", take that with a grain of salt.

You guys would be wise to get on the following runs: Cooper, Ingalls, Peshastin, Wenatchee playboating, Lower Icicle, Fall In the Wall, Skykomish (all in the Seattle/Leavenworth proximity). Going further north, you could hop on the NF Nooksack, and the Chilliwack just over the border. Also near that zone would be the Cascade and Ruby/Granite.

In hood river, you would have close access to the husum section of the white salmon and the farmlands section. Green Truss and the Little White would be out of range given what you have said about your skills as they are both difficult, class V runs. Beyond those runs, I don't know much else about the Hood River area, and there may be more to paddle around there, but you will probably have more different kayaking to do up north. You can always take a day or two to spend in Hood River.

Also, late may to late june would maximize the likelihood you have good water levels.

Hey P. We boated a couple times together when you lived in DC. Heard you moved south. Hope you're well. We did a similar trip mid June from out of town. Camped at the simple, but effective spot up at the Buck Creek trail head (find Northwestern Park near Husum and keep going on Buck Creek). Good location. Lots of boating nearby, but water level dependent might be hard pressed to fill a week. I would add the Klickitat was a nice Class 3 float (only to fish hatchery) and might be fun as a very simple overnighter if you wanted to do the length of it. It was pretty, not that far away, but was kinda different than the stuff near Husum. Also look into the Wind. I think it has a section to consider; possibly w/some portages. I think it was high when we were there so only some ran it. Also Hood River is supposed to be fun, but was too low on our trip.

Lots of other fun stuff to do in the Hood area. Good food, biking, fishing and great weather.
Not sure about Next Adventure rentals. They really screwed a friend who rented a boat. I would pursue other options first. We usually pick up boats in Seattle at Aqua Sports in Redmond. Have fun!