The comments made by Bradshaw in an interview with the Financial Times follow the BBC's rejection of government proposals to redirect a portion of the licence fee to funding the regional news production of commercial broadcasters.
They have also overshadowed the BBC Trust's announcement last night that bonuses for the 10 most senior BBC executives will be suspended.
Thompson and Lyons have argued that such a move will create a dangerous precedent which will endanger the BBC's future funding and its independence.
Thompson last month accused ministers of "ideological" motivation.
Bradshaw has criticised the pair by saying that their misguided stance on this issue is part of a "pattern" of poor leadership at the corporation.
Bradshaw said: "[There] are plenty of people within the BBC who do not feel it is a well-led organisation and that is almost for me the most worrying thing.
"And they don't feel they are being well-led on this issue. It fits into a pattern. It is not the only issue.
"There is almost a feeling of despair among a lot of highly respected BBC professionals."
He called on the BBC heads to "show some leadership" during the consultation period between now and September over the proposals, which call for 3.5% of the £3.6bn licence fee to be redirected.
Some argue that the money is coming out of the portion of the licence fee which was intended to be used to support digital television switchover between now and 2012.
Bradshaw, a former BBC journalist who was appointed culture secretary in a Cabinet reshuffle last month, said: "I think the BBC is far more likely to be able to make a strong case for the retention of the licence fee if it sees itself as an organisation that is not just simply always interested in defending its own narrow interests, but has a broader role in terms of defending and providing high quality public service content."
The culture secretary's broadside coincided with a move by the BBC Trust to curtail executive salaries in response to public feeling and the recession.
The Trust has suspended bonuses for the 10 most senior executives, who include Thompson and Jana Bennett, the director of BBC Vision.
This follows moves such as imposing a pay freeze for all managers earning £60,000 or more until July 2010 at the earliest.