During the initial process make sure that the superior of the person making the decision knows that their requirements are for a product that they were told has a bad reputation in terms of integration with standard administrative tools, performance, corruption, overhead for configuring, etc. If you can make that more widely known, all the better. But the boss is definitely the person who needs to be informed. After all the desire for a standard product from a known company with a known brand name is basically a CYA response, so make sure that the A is swinging naked in the breeze.
You will probably lose the contract. Don't sweat it and don't waste too much energy. That is a lost cause against entrenched assumptions and you know it.
Then every month or 3 send an email to the person who made the decision and that person's boss, asking them how the project is going, how the tool you warned them about is working out, reiterating your original estimate, and saying that while you are disappointed that you didn't get that contract, you would dearly love to get a chance to show what you can do.
Don't put a lot of work into this. (But if you can get inside information and time your emails to coincide with missed deadlines, major crashes, etc...alternately you could make the emails coincide with what you estimated as in your initial estimate as when you would hit major milestones.) If you are right about their current toolset's deficiencies, their tools will be doing the convincing for you. Your job is just to keep them from slipping into the apathetic belief that this is just how it has to be.
What you are aiming for is to turn those problems with their toolset into active dissatisfaction, so they are willing on a future project to give you a chance. If you get that chance, then do your best to bid low, estimate the time high, and then overdeliver. You got this chance through dirty politics and there are very likely to be people in the company with considerable investment with the current tools wanting you to fail. But if you succeed in proving yourself well...