VRE, unlike Amtrak and MARC, does not have dedicated bike racks. That said, I’ve not been hassled for not having a “bungee.”
Last time I checked it was a flat fee of $20 each way on Amtrak for bikes, and must be reserved at the same time you purchase the ticket. Almost as much as the rail fare. I have been toying with the idea of an Amtrak round-trip to Culpeper. The schedule gives you plenty of time to bang out at least a metric and then enjoy a nice meal before catching the evening train. A lot of nice, scenic routes around Culpeper.
Another thing I’ve been thinking about. On the MARC Brunswick line there is an early “getaway” train that runs Fridays only. Departs Union Station at 1:30. Ride to Dickerson, Point of Rocks, Brunswick, or similar get in a good ride back to DC. Would require playing hooky for a half-day. Doable in complete daylight in a few weeks, and especially after the time change in March.