Hop scotching across several items:
The term "wheelchair" is too limited, "Disabled" properly fits the bill.
Amtrak's (and generally the ADA regulations) definition to qualify for a disability fare and advance qualification for an "H" room is:
Reservation by telephone or at a ticket counter.
Sorry, discounts are not available when you book online.
You must also provide written documentation of disability at the ticket counter and when boarding the train.
Acceptable documentation includes:
Transit system ID card
Membership card from a disabilities organization
Letter from a physician
When Reservations Are Required
To ensure that you get the space and accommodations you require, you must make a reservation for each of the following:
Wheelchair space
Transfer seats (for when you travel in a seat and stow your wheelchair)
Accessible sleeper accommodations
We require that you make reservations for such accommodations on all trains, including on "unreserved trains" (on which reservations for ordinary seats are not required).
Accessible space is limited. Please make your reservation as far in advance of travel as possible.
Who May Reserve Accessible Accommodations
Up until 14 days prior to the departure of each train from its origin city, reservations for accessible bedrooms may be made only for passengers who are mobility impaired.
After this period, and if all other Deluxe and Family bedrooms have been reserved, accessible bedrooms are made available to all passengers on a first-come, first-served basis. For this reason, we urge you to make your reservations as far in advance of travel as possible.
For a party including a qualified person the H room accommodation fare is priced as a Roomette, a non-qualifying party pays Bedroom fare.
About Yield Management
Yield Management (so-called "Bucket System") does not require that any fare start at the lowest point.
Actually, "Experts" start at the top and look at trends, holidays, vacations, etc. and try to predict if any empty seats will exist at the peak fare.Remember the idea is to get the greatest revenue. If so, they look at how to fill those empty seats by price reductions. Although Amtrak uses fixed bucket levels, they have been known to offer last minute reductions - after all anything you can get for an otherwise empty seat or room is gravy.