I know how to do this now so I thought it would be best to revisit and answer my own question. As it happens, the authors of this study (https://doi.org/10.1093/gbe/evab209) do exactly this in their paper and have released all the scripts to reproduce their work, including simulating neutral evolution given background selection using SFS_CODE, on github here: https://github.com/Jimi92/Cerco-DMI-resistance
So reading the paper and these scripts should answer how to do this for anyone interested. But here's a quick example taken from the github (but I modified the output file name):
./sfs_code 1 2 -n 89 -t 0.0017 -P 1 -L 1 1000000 -Td 0 0.026 -TE 0.0047 -r 0.063 -W 1 200.0 0.0 1.0 -o output_file.txt
Not all of this is directly relevant to the background selection part (some of it specifies a specific demographic scenario, the mutation rate, number of samples, number of populations etc etc), and if you want to know the details of what all these flags mean, it's listed in the github (also in the documentation, but the github lays out the meaning of the specific flags used in this example very nicely).
But the most important part for simulating negative / background selection is the W flag, which specifies the distribution of selective effects. In this particular example the authors used -W 1 200.0 0.0 1.0. There are four values that come after the -W flag and the meaning of these terms is <type> <gamma> <p_pos> <p_neg>. To understand <type>, it's complicated and best to just consult the SFS_CODE documentation directly (on pg. 16 at the time I'm looking at it) https://sfscode.sourceforge.net/SFS_CODE_doc.pdf.
The <gamma> refers to the γ = 2 Ns value, representing the selection coefficient. And, to my understanding, <p_pos> and <p_neg> represent the percentage of new nonsynyonmous mutations which are advantageous, deleterious respectively. The documentation itself specifies: "Of course if you simply want a Γ-distribution of negative selection (assuming no positive selection), then you can simply set <p_pos> = 0" (pg. 17). For this reason, the above example I chosen from the authors scripts uses <p_pos> <p_neg> of 0 1.0.