The biggest issue is that to challenge them, you have to dox yourself, even if the DMCA claim is completely bogus and itself is contains bullshit information about nobody who actually exists (e.g, in a doxing war, the party that fires a bullshit DMCA claim first has a huge advantage).
You pay lawyer, they back off, you're now in the red for lawyer fees. I guess you could sue for damages but good fucking luck
1) Do you have to pay for the lawyer? Can't you object yourself?
2) Shouldn't offending side reimburse the expenses?
> Shouldn't offending side reimburse the expenses?
In an ideal world. They might even be legally liable in this one. But you still have to sue them to get the money, which is an expensive gamble for a very small pay off.