Assuming - from the wording of your opening post - that cleaning the existing carpet covering rather than replace with something new is your preferred solution:
Have you tried a more aggressive cleaner than you may already have been using? (Such as
http://www.homedepot.com/p/ZEP-1-gal-Industrial-Purple-Degreaser-ZU0856128/100047759 ). Applied to the whole carpet surface? With a garden sprayer so to "mist" over the carpet surface but not to over-saturate. Then scrubbed for a short while with a stiff brush? Then use a wet/dry vac to remove the excess liquid? Then repeat several times but with the sprayer now only filled fresh water? And do this only on a warm/hot low humidity sequence of days so that the whetted carpet can dry fully dry out before mold begins to grow again?
Might be worth a try before you do something more time demanding.
One item of warning about the above sequence. Industrial Purple right out of the jug before dilution with water actually is almost as aggressive as a water-based paint remover. So do your best not to over-spray onto adjacent finished wood work or let mist onto (say) your cabin floor. And get any cabin upholstery out of there! For good measure as you spray the carpet continually wipe/clean adjacent surfaces with a water saturated rag before any over-spray begins to cause collateral damage. Also, wear vinyl or latex gloves and not let linger on the skin. Make sure that your boat is well ventilated and that you have some sort of a respiratory mask on.
I mention Industrial Purple because my 1980 boat (a Hunter) when I bought it had OEM vinyl straw weave wall-paper on several interior surfaces. It was very badly stained after 25 years of humidity and mold. All references I found said the only solution was to remove. But when I wiped on a strong concentration of Industrial Purple and then cleaned off with wet rags, the original surface came back almost as if new. I also learned during this process that the Industrial Purple stuff did remove my teak's surface oil finish. Yours is a carpet rather than a vinyl wall-paper. But just an hour of work for a small area as above to see if it works is a good trade-off compared to the dozens/workweeks of hours that you will spend removing the current carpeting and finding something else to put in its place. Try first in just one segment.
Just my take ...