I would agree with your boss that this is ungrammatical.
Gerunds have verb-like properties and can be modified by adverbs, yes; but the placement of the adverb should in that case be parallel to the placement of the adverb in a corresponding verbal clause.
Some adverbs (can) have a difference in meaning depending on their placement; for example:
I just quickly went to the store. (= I was only gone for a minute)
I just went quickly to the store. (= I walked very fast)
Others have a preferred location, and moving the adverb makes the sentence ungrammatical (or at least very, very clunky and awkward):
We manage our time better by doing it this way.
?/*We better manage our time by doing it this way.
I nearly choked on my sandwich.
*I choked nearly on my sandwich.
The latter form with better here is perhaps marginally grammatical, but it is clunky. The latter form with nearly is quite ungrammatical.
When the verbal clause is transmorphed into a gerundic noun clause, the preference for a particular placing goes along, but the ‘weight’ of the preference in the verbal clause is strengthened, so that an adverb placement that in the verbal clause is unwieldy and clunky becomes downright ungrammatical in the noun phrase:
Quickly going to the store. (fine) / Going quickly to the store. (fine)
Managing our time better. (fine) / *Better managing our time. (clumsy to the point of ungrammaticality)
Nearly choking on my sandwich. (fine) / *Choking nearly on my sandwich. (quite ungrammatical)
At least, this is the case for me. It is possible that this is an area where different speakers have different levels of tolerance, and placements that in verbal clauses are only clunky, not ungrammatical, remain that way in transmorphed noun phrases.
As for your last question:
No, the gerund managing is neither subject nor object in either case. It is part of two noun phrases, of which the first is the subject (the verb being means), and the second is the subject complement in the same clause.