In my experience, one does not benefit that much from such an approach in this case (mammal vs bacterium). Besides, wouldn't you still have the false positive problem if you pre-mapped to pig? (i e wrongly assigning bacterial reads to pig) Anyway, I think there is enough divergence that you would not get many false positive hits.
I did such a project (mouse + a certain bacterium) and there the problem was rather that we had a bacterial genome that we mapped to and got a lot of hits to, but in the end it turned out that most of those were from other bacteria (mostly E coli) that were contaminating the sample. A useful way to spot such erroneous cross-bacterial mappings turned out to be looking at the "mismatch spectrum" i e how many of your reads have 0, 1, 2 ... mismatches in the alignment to the genome you are mapping against (this is easy to get from the BAM/SAM). It should be mostly zeroes and ones, but in our case the most common number of mismatches was 6 or so. So the bacteria were close enough that reads from one could be mapped to the other, but always with many mismatches. Perhaps that could be helpful for you as well.