Steven Lowe has some good thoughts. It is possible that you can work something out with the client.
Your job isn't to worry about how other contractors will interface with the code you gave your client, your job is to win the new contract if you decide to bid for it.
If you decide to bid, then the following applies:
Critical to the the process is how much of a chance to do you feel you have of winning? Some organizations put out an RFP even when they expect the current provider to win it because they have contracting rules or laws to follow. If this is the case, likely you still have a pretty good chance of winning. If you believe they did this because they were not happy with your team's current work and want a new group, you have a long way to go to win.
First rule of RFPs is to follow the directions exactly. If they say they want three pages don't give them four. In general when bidding (especially government bids), the bids which do not do everything as exactly as specified get thrown out unread.
Next job is to sell your company. Somewhere in the RFP is a section where you describe the experience of the company - make sure to strongly highlight your company's expertise in developing the system, same when you provide bios of the staff if that is required. In the actual proposed solution part of the RFP, you can use your inside knowledge of how the system currently works to show what you would do to enhance it. Etc. If the client has told you they are unhappy with your product as is or with the delivery of the product, try to include verbiage that shows how you will change to address past problems. I wouldn't do that if they have not expressed this to you but only if you feel that your company is unlikely to win for a specific, expressed reason. Then you need to do something to counteract that perception. For instance, if they felt that there wasn't enough testing, explain the new company testing policy and how it will be documented to the client. If they felt that you were unable to meet deadlines, try to develop a more realistic deadline in the proposal and show how you intend to manage the project to keep it on track.
As far as costing, your people won't have to get up-to-speed to understand the code. That means you should be able to show that you can do the enhancements in fewer hours than your competitors. This is a big plus for your side. Find a way to make that point somewhere in the bid.
Now as far as the maintenance and coordination fo code bases if you do not win the enhancements, that is something you need to work out with the client. If the source code is thiers (which it sounds as if it is), then it is up to them to figure out how to manage it between two vendors. Be cooperative, but it isn't your job to solve that problem for them only to provide the code as agreed to in you current contract. Certainly both groups would need to have source control and that should probably be a client-maintained repository that either group can access. Ask for aplan as to how to determine if something is part of the enhancement contract or system maintenance. It's easy at first, but not so easy the week after the first enhancement goes live. Ask them how they intend to get your maintenance people familiar with the enhancements that they will need to support.
How professionally you behave if you lose the enhancements but continue the maintenance might have a lot of bearing on future work. Graciously giving the other team what they need to do their job will serve you better than being pissy about it. And don't be surprised if the other team hires some of your developers away from you if they win. Likely you will need fewer developers if you lose and, it being a small world, again I would let those people go without fighting it unless you have a really strict non-compete clause in their contracts (even then I would tend to let it go in this case if you lost the contract).