If I bought my own receiver, would they still charge me $5 a month to use it?
Yes, actually it's $4.99 for additional non-leased receivers.
Most of what I'm going to tell you is going to be dependent on your installer. If you can't get them to work with any of these options, you might be stuck for the moment.
You do have another, albiet pricey, option. You can get a dpp44 switch and two separators, put the switch in your wiring closet, use a single line from it to each receiver, connect the line to the separator input and connect the two separator outputs to the two sat inputs on the receivers. That free's you up for using the second line that you ran for the receiver output back to the closet and wherever you want to go from there. You will have two empty ports off the dpp44 for additional receivers. This option will probably cost you around $250.
You can also use diplexors to send the receiver output back up one of the two lines to the closet and then wherever you want it, however this does not address your question about an additional receiver. Most likely you will be getting a dishpro quad lnb unless your locals are not in the 110 or 119 slots. That quad will only support 4 tuners, which is what you will have. You will not be able to add another receiver without changing your setup to include a switch. Diplexors (good ones) generally run $10 to $20 each, and you need them in pairs for this application. You also have to convice the installer to put them in if you get them, or not hook up the 2nd receiver out and you do it yourself later.
If your locals are NOT in the 110 or 119 slots, the install will include a switch, most likely a dp34. If you can work some kind of deal with the installer to pay extra and get a dpp44 I would do that, otherwise you'll have to decide what you want to do. You can stick with the diplexing option, purchase a second dp34 and cascade it off the first one at a later date when you get the additional receiver, or you can bite the bullet and do everything with your existing wiring, a dpp44 and two separators.
If you DO go with a dpp44, there may be one temporary catch at the beginning. Depending on the software on the receivers when you get them, you may need to temporarily hook up both lines directly to separate ports on the switch so they can download the latest software. Once that is done, you can go to the single feed, seperated at the receiver.