You can't imply additional work into a contract. Work content is normally specified and whilst "associated" works may be included, additional works certainly are not.
Contract terms have to be reasonable and fair. No clause that allows to vary costs unconditionally is fair, and so not enforceable. The only time such a condition would be reasonable is in situations where the work content is unknown, and even then additional work and additional costs must be agreed beforehand.
The BCO is not paying for the work so can't tell anyone what to do.
You will find it difficult going elsewhere for quotes or even getting the work done - what builder wants to go on a job with another builder working there? Even if you tried for quotes, any builder would be within his rights to hit you with claims for delays and time as you are preventing him from carrying on with his work and meeting his other commitments to you and others.
You seem to have numerous problems and made all the common mistakes - a crap designer, a crap or no specification, no clear detailing of the work or work content, no mutually agreed contract terms, and most significantly no provision by which additional works are quantified, valued or agreed. The end result is that the builder has you by the balls and does what he wants with your money and there is little you can do because ... you've actually agreed to it!