No, the Merriam-Webster example is not wrong.
As far as Fowler, because he is a prescriptivist, I'm not sure it makes sense to call him 'wrong'. So he's advocating a ban on gerund-participials after as well as. Well, all I can say is that such a blanket ban would go against very solidly established usage, and that there is nothing intrinsically ungrammatical about the usage he wishes to ban. Style, however, is a different matter. I do agree that his gerund-free rewritings of the sample sentences do sound better than the originals, to my ear at least. But the originals are all perfectly acceptable English.
The three functions of as well as
The point is that as well as has three uses, of which one is not relevant to your question, while the other two are.
The one that's not relevant to you is its literal meaning, which is used in comparisons of equality (He played as well as he'd ever done).
Now onto the two relevant meanings, both of which we may call idiomatic.
As well as functioning as a coordinator
In its first relevant meaning, as well as is reanalyzed as a coordinator that roughly means and. One slight difference when using as well as this way is that, unlike with and, the second coordinate gets backgrounded. This as well as has most of the properties typical of coordinators. Just as with and, or, etc., the coordinates joined by as well as normally have to be syntactically alike (i.e if one is a noun phrase (NP), the other one must be as well; if one is a verb phrase (VP), the other one must be as well; etc.). And just like with and, if it's a coordination of NPs which serves as the subject of a clause, then the corresponding verb has to have plural agreement. An example of this is the were here: [Abstraction] [as well as impressionism] were Russian inventions.
This 'coordinator' meaning is the one used in the Merriam-Webster example. That example is a coordination of VPs. Note that as well as could be replaced by and without losing grammaticality, and with a minimal change in meaning, if any:
[We offer electronic toys] as well as [rent out video games].
[We offer electronic toys] and [rent out video games].
As well as functioning to introduce a subordinate element
In its second relevant meaning, as well as functions differently. In this case, it introduces an element that is subordinate rather than coordinate. Therefore the elements that this as well as joins need not be syntactically alike, which is one of the reasons why we wouldn't be able to substitute and for as well as in such cases. This is the use of as well as seen in examples such as Smoking is dangerous, as well as making you smell bad.
Analysis from CGEL
For more details, here is CGEL (pp. 1316-1317):
The literal use of as well as is seen in comparisons of equality like He played as well as he'd ever done. Here well is an adverb heading the underlined [boldfaced] phrase, an adjunct of manner. There is also an idiomatic use meaning approximately "and, in addition to", illustrated in:
 i a. She [means what she says] [as well as says what she means].
b. [Abstraction] [as well as impressionism] were Russian inventions.
c. [Both increasing ewe liveweight,] [as well as liveweightat mating,] influence
ovulation rate and lambing performance.
ii a. [Beauty] [as well as love] is redemptive.
b. He will have, [as well as the TV stations,] [a book publishing empire].
c. I met her father, [whom] she had invited along [as well as her college friends].
d. She [has experience in management], [as well as being an actor of talent].
In [i] as well as behaves like the coordinator and. In [ia] it links two finite VPs (verb phrases), a property characteristic of coordinators: cf. property (c) of §2.1. Note in this connection that while She plays the piano as well as the violin (with paired NPs) is ambiguous between a literal meaning ("as proficiently") and the idiomatic one ("and"), She plays the piano as well as sings lieder (with paired finite VPs) has only the idiomatic meaning. In [ib] the form were indicates that the subject NP is plural, just like abstraction and impressionism. And in [ic] we have not only such plural agreement, but also a correlative pairing of both with as well as instead of the usual and.
In [70ii], by contrast, as well as behaves markedly differently from a coordinator. In [iia] the 3rd person singular verb-form is indicates that this time the subject is singular: is agrees with beauty, so that as well as love is treated syntactically as an adjunct, not a coordinate. In [iib] as well as the TV stations precedes a book publishing empire, making it clearly an adjunct. And could not appear in the position as well as has here: cf. property (d) of §2.1. In [iic] relativisation has applied to just one of the bracketed constituents, contrary to coordinator property (e). And in [iid] the bracketed constituents are syntactically unlike, the first being a finite VP, the second a gerund-participial, contrary to coordinator property (b). Note that order reversal is possible in [iid] (As well as being an actor of talent, she has experience of management), but not in [ia] (*As well as says what she means, she means what she says).
We must conclude that idiomatic as well as can be construed syntactically in two ways, introducing an element that is either coordinate (as in [70i]) or subordinate (as in [ii]). In the former case, we take it to have been reanalysed as a compound coordinator. In the latter case there has been no such syntactic reanalysis, and here as well as does not form a constituent. This is evident from the fact that as well can occur on its own: compare Beauty is redemptive and love is as well In [iia], then, the second as is a preposition taking the NP love as its complement, and the whole PP as love is an indirect complement in the AdvP as well as love. Similarly for the other examples in [ii].
As a coordinator, as well as is restricted to subclausal coordination: She plays the piano as well as she sings lieder, for example, has only the literal comparative interpretation. Even as a coordination, 'X as well as Y' differs from 'X and Y' in that the second term is backgrounded: Y often expresses information that is discourse-old, i.e. familiar from the prior discourse.