You can absolutely do this. I use a similar method to swap out a 20nt chunk of a plasmid routinely. Typically, I add excess of the 5' and 3' primers and roughly equivalent molar amounts of each template and each single-stranded 60nt oligo. The oligo has ~30bp homology to the other oligo and ~30bp homology to the template. It looks (roughly) like this:

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In your case, simply make your primer such that it includes ~20bp homology upstream of your deletion, and then ~20bp downstream of it (the annealing temperatures of your primers to the double-stranded template and the other single-stranded primer should be similar.) Keep in mind that, depending on your primer design, you may end up with PCR artifacts (multiple bands) requiring gel extraction, and your PCR may be low efficiency. I use 3% DMSO in my assembly PCR, with an annealing temperature ~3C lower than calculated, and a generous extension time.

Alternately, I've also tried tossing all four pieces with a digested plasmid backbone into a Gibson assembly (5 piece Gibson with 3 double-stranded and two single-stranded pieces) and I've gotten successful assembly and transformation in several cases. Definitely sequence-verify your results before proceeding.

sorry I could not help because it is not my interest

i agree with Sarah Stainbrook

You are definitely able to do so. 

Good luck