If you want to add to your physical training I'd say you should do some form of high quality resistance training, which generally means lifting weights though there are some other options. It is by far the most efficient and safest way to get stronger and look fitter. Calisthenics, aerobics, and most martial arts classes aren't going to improve your strength much by comparison. You'll also find that you really benefit from the extra strength. Most everything you do in your MA training will be easier and building muscle will help protect your joints and the rest of your body from injury. This can be particularly valuable while you are carrying a lot of extra weight if you are engaged in a martial art that involves a lot of jumping or other activities that create a lot of impact on your knees. If your goal is just to look fit, putting on some muscle is still one of the fastest ways to achieve that.
I know you said you don't want to look like a body builder and so you may be put off by the idea of lifting weights. The question to ask yourself is, were you too muscular looking when you were 18? If not then it's extremely unlikely you've got the genetics to suddenly look like Brandon Curry if you add in a couple of strength training sessions a week at the age of 40. To get body builder kind of big, or even sort of close to it, takes some combination of genetics, dedication and in the case of many (all?) of the pro's a willingness to use controlled substances. It's not something you have to worry about happening by accident.
As many others have said, fat loss is going to be most impacted by changes in eating habits. Exercise will help but is very unlikely to be sufficient to reach the goals you're talking about alone. The only thing I'd add on this point is that weight by itself is a very poor metric to use to measure progress. If you put on 10 lbs. of muscle and lose 5 lbs. of fat you are making great strides towards you goals but it will look like you're failing miserably and moving backwards if you only look at the scale. Body composition is what matters, not weight.
If you're serious about making big changes in your appearance I recommend choosing one of two approaches to measure your progress; 1) using a soft set of metrics, asking yourself questions like, "are my shirts starting to be looser around the belly and tighter in the shoulders? Is my belt a notch looser or a notch tighter than it was last month?" and ignore the scale, or 2) If you want to have a more detailed picture of your progress go full data geek and make sure you've got good information to work with. Get a DEXA scan or do hydrostatic testing at regular intervals and take measurements and maybe use calipers along with the scale in between. Either can work, just pick the one that you find motivating rather than demotivating and skip the scale altogether unless you combine it with other measurements.