Is there any record of how many starships the Federation lost in battle with the Dominion forces during the war?
There is no canonical answer to your question.
No television episode of DS9, no episode of any other Star Trek series, and no Star Trek film has commented in an exact way upon loss of hardware (or lives) by the Federation in the Dominion War. (I have checked scripts.) None of the writers of DS9 have discussed Federation losses in interviews or other commentary. (Again, I have checked.)
Memory Alpha ("Dominion War") only describes casualties of the Federation-Klingon-Romulan alliance as "severe". We can assume, then, that many ships were lost, but any attempt at counting the loss of ships would result in pure speculation.
The only numerical information canonically associated to the war is the loss of Cardassian lives, estimated in the series finale of DS9 to be approximately 807 million. (See the same Memory Alpha article.)
You may find the following thread useful, but most of it is pure supposition:
There is one theoretical method but it has too much uncertainty.
In one episodes the augmented humans calculated that the Federation would lose the war and six hundred billion citizens - military and civilian - would be killed.
It is hoped that the war was ended while the actual casualties were still only a tiny fraction of that estimate.
If, for example, ten percent of the predicted dead were military and ten percent of them were ship crews, then six million members of ship crews would be killed. Dividing that by the average size of a ship crew would give the number of predicted ships destroyed.
If in this example the average crew size was 600, 10,000 ships would be predicted to be destroyed.
Dividing the number of ships predicted to be destroyed by the ratio of predicted deaths to actual deaths would give the actual number of ships killed.
That method obviously has too much uncertainty.