Is this usage of "lest" possible?

The only danger is, lest it should be Too strong a remedy ...Source

Can this "lest" be used in this way?

Like this...

My apprehension is lest my advice should kill her motivation.

Not as such, really. The word "lest" can be loosely read as "if" or "in case".

Let's have a look at a larger excerpt of your source text, reformatted more prosaically:

The only danger is, lest it should be too strong a remedy; lest, in removing cold, it should beget too violent a heat; and into madness turn the lethargy.

As you can see, the two clauses I highlighted above indicate possible reasons that taking the action ("removing cold") could result in the stated outcome ("into madness turn the lethargy").

Thus, your sentence could be better expressed as

My apprehension is {great/well-founded/justified/etc.,} lest my advice should kill her motivation.

The addition of an adjective and a comma fixes it right up. (I can't explain why I struck out the "should", but my native-English-speaker "Spidey-sense" tells me to get rid of it. I leave it as an exercise to the resident English majors to rationalize it. :-)