Edit per comments and updates:
Given that there was no Notice Period specified it is ok to offer a Notice Period, even though you are not bound to give one.
The "default" period is 2 weeks, but if you know you won't be able to fulfill such period I suggest you offer them one that you will be able to achieve (the 1.5 weeks).
Have in mind that you are under an at-will employment, so chances are they might terminate you right there the moment you hand your notice, and thus won't have to serve it.
Is this a bad thing, or an integrity issue on my part?
Yes I fear this may be negative for you.
You contract specifies an exact amount of days you should serve as Notice Period. You should serve those days up to the number specified in your contract, not less.
The company may decide to terminate you right there without having to serve the Notice Period, though, but in case they want you to serve it you are bound to sever the full time.