In the letter, Mr Bailey described parts of Dr Pester's evidence to the committee in May as "an optimistic view" and said "greater caution would have made sense".
The head of the Financial Conduct Authority (FCA) said in a letter to Treasury Committee chairwoman Nicky Morgan that the regulator would carry out a joint investigation with the Prudential Regulation Authority.
TSB has failed to pay compensation to customers who have been victims of fraud within two days of becoming aware of the problem, which is required under payment services regulations, Mr Bailey said in a letter to Nicky Morgan, chairwoman of the Commons Treasury select committee.
"The committee has lost confidence in his ability to provide a full and frank assessment of the problems at TSB, and to deal with them in the best interests of its customers".
Up to 1.9 million of TSB's 5.2 million customers began experiencing problems with online and mobile banking services on 23 April after accounts were migrated from an IT system inherited from the previous owner, Lloyds Banking Group, to Sabadell's technology operation.
"If he continues in his position, this could damage trust not only in TSB, but in the retail banking sector as a whole", the letter said.
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Nearly 2 million people - more than a third of TSB's 5.2 million customers - experienced problems when making mobile and online payments as the result of a software failure, and the problems have not been entirely resolved.
Paul Pester, the under-fire chief of crisis-hit bank TSB, is facing an unprecedented attempt by MPs to oust him, after the Treasury committee wrote to the bank's board demanding it reconsider his position as "a matter of urgency".
TSB Chairman Richard Meddings said the bank will publish its own review into what happened, being conducted by law firm Slaughter & May. I am deeply concerned by TSB's poor communications about the scale and nature of the problems it has faced; by its response to customer fraud; and by the quality and accuracy of the oral and written evidence provided by Dr Pester to the Committee.
TSB was not immediately available for comment.
Paul Pester told MPs on Wednesday that the bank experienced 70 times the normal level of fraud attacks last month after a new IT system was introduced in April which some customers struggled to use.
"We recognise that we still have areas where we need to improve performance for our customers and we do not underestimate these remaining issues".