Would there be a problem to date someone of a higher rank from a completely different chain of command?

Nope, not a problem at all as they are not in the same command and neither reports to or works for the other. Back in '93 we had a Marine Lance Corporal married to an Army Staff Sergeant. The hard part was the logistics but luckily both were assigned to postings in the DC area so it worked out. The problem comes in when they have to PCS to different parts of the globe. Can't exactly explain away a grunt on a Destroyer.

You can find this in the DA Pam 600-35, which you should follow up with by checking on yourself.
Basically, since you are a reservist you may date someone of a higher rank as long as your primary relationship with them is not through work. Being in a different unit would be fine. This only applies to Army. For instance, Air National Guard doesn't have this reserve exception. If you dated an officer from the Air National Guard they could be in trouble with their own chain of command.

My friend who was a SN (E-3) dated an Army Maj. It is ok as long as the Senior cannot influence the career of the junior or exercise control over them. Be discreet when both are in uniform so as to not give the appearance of fraternization.