Unless you can prove every cent of these fees were related to these lawsuits, you can't really say Obama's spent $2 million to hide his birth certificate.
And your contention cannot be proved that NO money went to Perkins & Coie that defended Obama in his eligibility cases, which is absurd on the face of it.
It's hardly necessary to account for every penny. The absurdity of the claim can be demonstrated with public information and basic math.
If Obama's been paying Perkins Coie to defend against Birther suits, then obviously Perkins Coie attorneys must have entered appearances and filed documents in the cases they've defended. These documents are public record.
Now, while there have been at least 67 Birther suits brought, Perkins Coie attorneys did not participate in all of them, or even in a majority of them. Some were summarily dismissed without an Answer even being filed. Many were handled by state attorneys (e.g., Ankeny v. Daniels). Many more were handled by U.S. attorneys.
No case has survived past a Motion to Dismiss. Thus, the most any counsel has had to do in defending a single Birther suit is little more than 1) file an Answer, 2) file a Motion to Dismiss and Brief, and 3) attend a hearing or two. There's been some variation, but Birther suits have not been time-intensive ordeals for defense counsel.
That said, how many suits have Perkins Coie attorneys actually defended Obama in? A quick search turned up several of the bigger cases that do NOT appear to have had Perkins Coie counsel (i.e., Ankeny, Barnett v. Obama, Cook v. Simtech, Kerchner v. Obama, Taitz v. Obama).
Meanwhile, I found only two cases (notable or otherwise) that DID have counsel from Perkins Coie: Berg v. Obama, and Hollister v. Soetoro. (Both, incidentally, were filed before the inauguration and both have Obama himself as the primary defendant. I believe this may be common to all Perkins Coie Birther cases, but we'll need other examples to be certain.)
So that's just two confirmed cases where Perkins Coie represented Obama. Let's be really generous and say that there are another eight that I missed, for a total of ten cases.
Ten cases into $2 million is $200,000 PER SUIT. Given the minimal amount of work involved in getting each suit dismissed, let's be extraordinarily generous and say they billed 50 hours per case. That's $10,000 per hour.
For instance, in Hollister, Bauer appears to have filed only four documents in the trial court: an entry of appearance, an 8-page motion to dismiss and brief, a 3-page reply to plaintiff's opposition to the motion to dismiss, and a motion to substitute counsel. All in all, less than 20 pages of filed documents. The "Obama spent $2 million" folks would then have you believe that Obama must have averaged $10,000 PER PAGE. For pages that could have easily been written by a first-year associate and a paralegal. (And remember, that's still assuming there are another 8 cases as yet unidentified; if there are fewer than 8, that $10,000 goes UP.)
This may be why you never see Birthers actually specify which cases were Perkins Coie cases; because to do so would immediately indicate how patently ridiculous the $2 million claim really is.