As I am sure you have already realized, this genomic island is likely one that has been subject to lateral gene transfer (also known as horizontal gene transfer). It is in effect foreign DNA incorporated into the genome, which is why the GC content differs from the rest of the genome. The fact that the region contains a transposase makes this even more obvious.
By running BLASTN and BLASTX, you have already done the most obvious analysis. I would probably also run a BLASTP with the translation products of the called genes just to be sure, although I do not expect that to reveal anything new. Also, just to be sure, I would them all through Pfam to search for domains in case the NCBI annotation pipeline missed something, bug again I do not have high expectations.
Except from the transposase, you are thus likely stuck with a bunch of hypothetical protein-coding genes with no homology to things sequenced before. So what can you do with them?
The first thing I would do would be to make a BLAST database of the genes within the region and use BLASTP to do all-vs-all search of the proteins within the region. You may find that although they are not similar to things sequenced elsewhere, some of them are duplicates of each other.
You can also run a number of sequence-based prediction tools over the sequences to characterize them. Do they look like they may have a signal peptide and be secreted proteins (e.g. SignalP)? Do they look like they might be transmembrane proteins (e.g. TMHMM)? Or might they be intrinsically disordered proteins (e.g. IUPred)? Do they contain low-complexity regions (which might get masked in BLAST searches and thus why you get no matches)?
Beyond that, you can look at simple statistical characterization of them: protein length distribution, amino acid composition etc.
For inspiration, you may want to have a look at this old paper of mine, where we did similar analyses:
Analysis of two large functionally uncharacterized regions in the Methanopyrus kandleri AV19 genome
You must have a specific interest in this Pseudomonas. Is that genomic island contributing to that interest (by any chance or you know otherwise already)? You could try to look to see what happens by deleting the island for one.
But before you get deep into investigating this you are sure that it is actually present (i.e. you can PCR it (or parts of it) etc) from the strain and that it is not a contaminant that got introduced in some step of the sequencing process.
It's there for sure :-)
I am doing the bioinf analysis for some lab people. So we first want to have more hints what it could be. Based on that there might be follow-up experiments.
Have you (or have you not) excluded the possibility of it being related to the original interest the lab had in this strain? If it is not related to that interest then you may be embarking on a goose chase :)
Check out the EDIT in the original question :-)