soliciting (solicitation) of customers through/by the competition

I believe this term could be used within your context, especially in the phrase "Abwerbung durch" / "soliciting (solicitation) by the competition"
although to solicit is of course also used for "anwerben".
But its negative connotation might be what you are looking for here.
If it's too strong, then maybe something like

competitors' client pitch
competitors' advertising (campaigns)
client (customer) enticements by other ...competitors
special client offers made by other ...competitors
advertising (campaigns) by other well known competitors

Abwerbung is an active noun if you will, and abwerben is a particular activity, carried out through, for example, special offers which is expressed in the "werbung"/advertisement" part of the word. Here, the "ab" part brings in the negative connotation. I believe that should be expressed as such in English as well.

It's the competitor who's doing the wooing, the winning (them) over.
I think it is mostly possible to use such terms by rephrasing the sentence, see below. But this is more difficult for "Abwerbung durch xxx" - maybe not quite a loss, but the attempt to steal/take away customers (but these might be too strong).
"Abwerbung" can be done by advertising. In addition, maybe all is not lost yet. It could possibly just be the attempt to garner customers.

a few more options:
within a longer sentence:
the competition's attempt to reach for / woo / win over our customers
The competitor tries to entice clients (away)

for "Abwerbung durch xxx":

enticement (of clients) through the competition / through other well-known competitors

http://www.oneclub.org/oc/education/clientpitch/
client pitch