Freeway Airport (I learned to fly there) is much nicer and more active and is the closest to the stadium.
I second Freeway, although I wouldn't include the term "nicer" in any of my description. Adequate is the description I would use. I never had any issues parking as a transient for a few days at a time on numerous occasions, but sometimes it took a little cajoling to use a vacant tie down spot or I just parked in the grass.
I always had someone pick me up, but suspect you could get an Enterprise car delivered or maybe Uber. Being on the east side makes it much more convenient to FedEx Field. On numerous occasions I drove up to Bowie, MD, from the Shenandoah Valley area and it always took at least an hour, usually more, to drive from the Dulles area on the west side to Bowie on the east side.
The security at Freeway doesn't amount to much. It's basically a knee high chain draped fence running from post to post. Maybe the sheriff deputy's car that was occasionally parked there helped with deterrence a bit.
The locals used to tell me about how the TSA would occasionally drive in for an inspection and leave just shaking their head. I guess as a privately owned facility, the TSA couldn't really force any fencing or security improvements.
I always enjoyed flying in there because it was convenient and I felt like I was poking the bear. With the exception of the SFRA requirements, which aren't that big of a deal, and the airport only 7 nm from Andrews AFB, Freeway has all the ambience of a country airport in the middle of Kansas or Nebraska, albeit a little busier. As much as the FAA and TSA have pontificated about security, there isn't much they can do to change it.
Tipton/Fort Meade Airport, also on the east side is a viable option, although it would be further away from FedEx. It is more like a typical GA airport with a 8' perimeter security fence and plenty of transient parking on a big concrete ramp.
If you want to grease your path flying into the area, file IFR and the red carpet will be rolled out for you. If you ever squawk standby or turn your transponder off, expect a visit from a Blackhawk helicopter and a license suspension, neither of which I've ever experienced in maybe 30 or 40 trips. One time I was cleared for and flew an approach in IMC conditions directly over the top of BWI. Then cancelled with BWI tower when I had Tipton in sight.