Bogdanovic, three years Murray's senior, has had a more troubled time under Britain's Davis Cup umbrella.
Wild card Bogdanovic, 23, is ranked 70 places below 13th seed Lee at 117th in the world.
Bogdanovic admitted he does not deserve another wildcard and told BBC Sport he would focus his career on doubles.
Bogdanovic will meet the winner of the match between Andy Murray and Tim Henman which takes place on Wednesday.
"I just tried to stay as aggressive as possible, " Bogdanovic told BBC Sport.
Fellow Britons Alex Bogdanovic, Laura Robson and Naomi Cavaday are all still to play because of a rain delay.
Doubles partner Ward, the British number four in singles, defended Bogdanovic and insisted the focus on Wimbledon results is unfair.
Bogdanovic recovered from losing the first set to level the match and was a break down in the final set.
Jamie Baker and Alex Bogdanovic both lost in their first-round matches on Tuesday, while Greg Rusedski and Martin Lee were defeated the day before.
The pair will meet in round two after Bogdanovic beat former semi-finalist Hyung-Taik Lee 6-4 6-4, and Baker saw off Alexander Peya 3-6 7-6 (7-5) 6-3.
Baker went down 6-1 6-3 to in-form American Mardy Fish in his first ATP Tour event and Bogdanovic lost 7-6 6-4 to experienced Russian Dmitry Tursunov.
The absence of both Murray and Bogdanovic is very likely to mean a place in the singles squad for Scotland's Jamie Baker, plus Dan Evans and James Ward.
"If he was just working for the LTA, I see that as a great opportunity for Alex Bogdanovic or Jamie Baker to have Brad full time, " Petchey told BBC Sport.
Britain's next opponents, Israel, are hardly likely to be quaking in their boots ahead of the 4-6 March match against a likely trio of Bogdanovic, Murray and the 187th-ranked Arvind Parmar.
Annacone, the LTA's head coach of men's tennis, was full of praise for Bogdanovic, who turned professional in 2002 and reached 108th in the world in June 2007, but is mystified by his lack of success.
Annacone described Bogdanovic as "one of the most puzzling guys I've been around" and also revealed the LTA may stop funding the Londoner if he does not make progress during the second half of the year.
"I probably won't get another wildcard and that's fair enough - I've had many chances, " said Bogdanovic, who partnered James Ward to victory over David Martin and Jean-Claude Scherrer in round one of the men's doubles.
Vedran Bogdanovic, a veterinarian and dairy farm manager of close to 400 cows in Kapelna, Croatia, says Farmeron helped him meet laws for animal tracking and sales and also for reporting animal deaths to insurance companies.
British number one Andy Murray has chosen to take a break from the Davis Cup while number two Alex Bogdanovic also opted not to take part in the tie, which is Leon Smith's first as captain.
While Isner was almost delirious, Mahut, 28 and ranked 148 in the world, looked remarkably fresh, perhaps boosted by the knowledge he had won a match with Britain's Alex Bogdanovic in qualifying 24-22 in the final set.
The team will now be without the top two players in Britain, as Alex Bogdanovic has also decided not to participate, having had his Lawn Tennis Association funding cut and been refused a wildcard into Wimbledon despite meeting the LTA criteria.
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