First of all, taking legal advice here (as in: the internets) is not a good idea.
Secondly, and this is just me speaking, not a lawyer, you should've thought of that before you took a LGPL'ed program and modified it for your employer.
If the license was something you could disregard just because you didn't like it, there wouldn't be much point in having one now would there?
If you and/or your employer is not willing to publish the source code with your modifications, you need to stop using that LGPL'ed code and get rid of it.
Again, that's just me speaking.
Get advice from a real lawyer.
In response to your question about circumventing the license by adding the code to a DLL, I would assume that would work in the following fashion.
What you would do would be to change the original program enough to make it able to call functions in external libraries. You would have to do so without making that piece of code specific to your needs, libraries, names of functions, etc.
Those changes you then publish, as per the license requirements.
Then you make your own external library with your own proprietary code and ask that program to load and execute it, using those modifications you made to it.
Not knowing the full extent of the LGPL license, I can't say if that is enough to avoid having to publish your library though I suspect it will.
However, again...
Get advice from a lawyer
The AGPL is based on the GPL, not the LGPL. It does not contain any linking exceptions, and any work using AGPL code (linked or otherwise, modified or not) must also be AGPL licensed and distributed.
Using separate processes can circumvent the (A)GPL, but this is murky ground. If your end application depends on the external process, such that it wouldn't function properly without it, then it would be considered a derived work of the AGPL software.
In most cases where people use separate GPL applications in closed source programs, they provide the GPL work as an optional extension, or an alternative back-end to some other piece of code etc.
The (A)GPL work cannot be distributed alongside the final application even as a separate app (eg, putting them into the same archive or repository), although it's fine to provide instructions on where to find the GPL work and how to use it with your app.
Best Answer
You never need to publish anything with the (A)GPL. You only ever need to make available the source code to those persons to whom you also make available a binary (and with the AGPL also to those users whom you let use your binary over the network).