Bug? Numerical calculation error with FullSimplify and arbitrary precision

I think I've traced down the problem. It hinges on two things. An identity:

Cosh[x] == Sinh[2 x]/(2 Sinh[x]) // Simplify
(*  True  *)

And a questionable auto-simplification:

Csch[0``10. n] Sinh[2 0``10. n]
(*  1  *)

{Csch[0``10. n], Sinh[2 0``10. n]} // FullForm
(*  List[Csch[Times[0``10.,n]],Sinh[Times[0``9.698970004336019,n]]]  *)

(The coefficients are equal, so I guess that's why they are treated as identical.)

Here are the steps in which these problems arise:

n Exp@(0``10 n) // ExpToTrig
(*  n (Cosh[0.*10^-10 n] + Sinh[0.*10^-10 n])  *)

n (Cosh[0``10. n] + Sinh[0``10. n]) //. SimplifyDump`CosToSinRules
(*  n (1/2 + Sinh[0.*10^-10 n])  *)

Sinh[0``10. n] // FullSimplify
(*  0  *)

The simplified expression is already wrong in the second step, in which the identity is applied. The last step is questionable, too, but understandable. It's what leads to n/2 as the answer.

I, for one, feel I understand why this bug has persisted.

Maybe the best workaround is to flush all "underflowed" arbitrary-precision numbers to exact 0:

FullSimplify[n E^(0``10 n) /. z_ /; z == 0 :> 0]
(*  n  *)