What is the minimum quantity of BTC that needs to be raised to make the project viable?
It depends on what level of the project you mean. There are the mission critical parts, like paying for bitcoin fees and notary nodes.
1000 would be enough for a reduced bitcoin notarization frequency and smaller number of notary nodes, but not a lot of development.
5000 would allow to have the fastest bitcoin notarization frequency, all 64 notary nodes and staff up a group of C coders and GUI devs fulltime for some years
10000 would allow having many LP nodes all fully funded for the lowest trading spreads and many liquid trading pairs
Above this would allow a warchest that can make opportunistic trades, arbitrages and other means to generate working capital and set up a very long term organization of the above. Along with some large scale marketing. Of course, even at the lower funds raised we will be doing marketing, but with a limited budget.