Have you any reason to doubt the pricing the QS gave you? Presumably as a paid professional with no vested interest in who does the work or what they might charge his / her costing is based on accepted industry standards.
Personally I would call a number of builders, give them the schedule of works (but not the QS's pricing) and ask for solid pricing and time-scales for completion. If a builder wants to charge for the pricing work, cross him off the list and look for another builder. You have already paid once for the pricing.
What part of the country are you in? There is little point in giving details of a builder in Cork if the house in Donegal, as he will more than likely want to do an on-site inspection.