Name correction in Passport for already initiated H1B

[Dharmendra Kumar]

Hi,

My name is Dharmendra Kumar

First thing is I have initiated H1B Visa for 2015 the petition is yet to be
filled by my Company. Second, My name in my passport is as
Given Name - Dharmendra Bijay
Surname - Kumar

But my actual names in all the certificates is as Dharmendra Kumar . No
middle name.

So the dilemma situation is when can I get the passport updated as I don't
have the new corrected passport while initiating my H1B visa so I have
initiated with the same passport (incorrect name) . If I go ahead and
change the passport now it will take 2 months approx to get the new
passport and my old passport will be cancelled by Indian embassy. So will
this be an issue while the document verification by US as I have given the
old passport no in the application which will be cancelled by the time the
time the lottery comes and petition sent for approval.

What should I do here?
Shall I do the name change in passport after the petition gets approved ?
or Shall I go ahead and get the name corrected , because I don't think I
will have enough time to get the name correction done after the petition
gets approved and before visa interview.

Please Help.