Make your gas giant not quite so giant.
Scale down the mass and size of your gas giant. The density of the gas giant is due to the large gravity of the high mass object. Or, depending on how you want to look at it, is due to the weight of all the extra mass higher in the atmosphere crushing down on it.
Scale your planet down so that there is less atmosphere to weight down on and squeeze it all into a dense form.
Eventually, if scaled down far enough, the gas will not hold together and will disperse.
Check out the astronomy.SE question Can small gas planets exist? Then make your gas planet as small as possible.
And add heat, as you mentioned.
To make this even better, you could, as you suggested, also add heat at the same time. Getting heat from a star is the easiest way to think of, but I just had another idea. Perhaps the gas planet is early in its formation (but will still hold this form for a long time), and it is in the process of clearing the matter in its orbit. If it attracts flammable materials, they could ignite within your planet to keep it heated.
There are quite a few ways you could claim there are flammable materials available in your planet's orbital path. The obvious ones would be recent nearby supernova. But perhaps there were previously (a) planet(s) nearby that have been destroyed, or maybe they still exist and are ejecting material that is being absorbed by your planet.
It is a bit of a stretch, but you can probably count on a reasonable amount of suspension of disbelief in your audience if you say your planet has a moon which ejects oxygen from geysers or volcanoes which escape the moon's orbit and are captured by the planet. A cloud of oxygen descending from the moon onto the gas planet could then be involved in a reaction that helps to maintain heat. Perhaps it settles to the center since it is lighter than the other elements and a few rocks at the center make occasional sparks, combined with the oxygen and some other element you get heat.
With the previous paragraph, you do need to bank on the audience not wondering "Why isn't that oxygen bonding with the hydrogen before that point?" You could probably use other chemicals instead which would provide a better and more solid excuse for you, but I'm not a chemist. You could also just not explain it fully and leave it at "Elements ejected from the gas planet's moons and attracted to the planet provide a slow, steady exothermic reaction which sustains the planet's temperature and density. This will not last forever, as the material in the moons is not infinite, but this situation will last many lifetimes." This is slightly hand-wavy, but it is not unreasonable and might be possible.
With the reduced mass this temperature increase might not need to be severe. So with the mass/size reduction combined with temperature increase via chemical acquisition leading to gradual endothermic processes, you should be good.