Tell the project bloke you'll send him on a holiday if he pushes your bid through as the preferred one. Usually works.
You tend to find unless you know of the firm managing the project, they'll often just get a bunch of other quotes for the customer or for their records in case it ever got questioned in the future, but they've often always got a firm in mind already. They just need say 5 tenders in on the table and they push the one they're mates with forward.
Obviously not all like that. But I worked with a bloke selling garden furniture, and he was an architectural engineer by trade and ran a building business with his old man back in the day. He actually built the BAL training centre (they dont use that one now mind) and when I was chatting to him about it, he said that in most cases he'd need to send the project manager who's managing the tenders a very nice drink and he'd often get the work.
He said if he knew the bloke who asked him to tender, he'd normally always get it too, and they'd discuss and acceptable price well before the tender got prepared.
Even councils have been done for doing it in the past. Letting tenders win for £1,000,000 when they had tenders in for £500,000 - that sort of thing. Stoke on Trent council got clocked doing it a few times, and they just kept saying the highest tender was the best one. Until one of the most reputable building firms around here butted in and said their tender went in and never won at half the price of the main firm.
They now use Keir based down south somewhere for the council work. When they said they'd use locals to support the economy. I wonder why they went with Keir?!?!?! (backhanderssssssss!!)