Picture Lake was mentioned in a comment. According to the USDA Forest Service, swimming and boating is not permitted on Picture Lake (my emphasis)
Picture Lake is the centerpiece of a strikingly beautiful landscape in
the Heather Meadows area.
Hikers only. No camping. No fires, except self-contained, carry-in
stoves. No snowmobiles. Dogs must be leashed within the developed
recreation area of Heather Meadows. Swimming and boating prohibited.
Do not enter areas closed for rehabilitation.
I'd guess that you'd need to consider your SUP on a case by case basis as determined by the Forest Service. However I think your best bet for identifying lakes where you can SUP is to find those that have a boat ramp.