Can the HED Tomcat wheels be made stronger?
When building a wheel, there are two main things the builder does to prevent spoke breakage: set the spoke line at the hub flange and rim when the spokes have little or no tension, and stress relieve the spokes once they're at nearly full tension.

Setting the spoke line means to cold set the material into the final shape it would otherwise be pulled into by the tension on it. This reduces fatigue, which is the most common cause of breakage. It also helps keep the wheel true. 

To do it, ideally the spokes would have little or no tension. Jobst Brandt in "The Bicycle Wheel" favors doing it with a little tension. I find I can tell what I'm doing better if they're pretty slack. At the hub end, push  each outside spoke hard against its flange. You're trying to get them to lay flat against the flange when not under tension, as opposed to bowing out. (There are other techniques, but doing it with your fingers works fine). At the rim/nipple end, you're trying to prevent any bowing where the spoke enters the nipple. Many wheels don't need any attention here, i.e. the rim hole allows the nipple to swivel naturally into an angle that allows the spoke to run straight, but for the ones that do, just grab pairs of consecutive pushing and pulling spokes that make a "V" (the intersection of the V being the spoke crossing nearest to the rim) and squeeze them together hard to cold set the spoke. I like to do this once the wheel is near its final tension, so that there's only minimal thread movement left to go and you're not pulling the section you just bent up into the nipple.

Stress relieving is a process of temporarily overloading the material to eliminate residual stresses left in the spoke elbows. Most breakage is caused by these stresses putting the elbow area above its fatigue limit in use as the wheel rolls along, each spoke getting loaded and unloaded. To stress relieve the spokes on a cross-laced wheel, find pairs of nearly parallel pushing and pulling spokes on each side of the wheel and squeeze them together hard, as though you're trying to break them. (Brandt's terminology). I would recommend doing this at around 80-90% of final spoke tension.

Because your spokes are having fatigue problems already, these steps are no guarantee because they can't fix fatigue that's already accumulated. But if there's a way of getting the wheel to stop breaking spokes without replacing all of them, this is it. Also, make sure the final tension is nice and high.

In the end these wheels are probably not strong enough to be trouble-free for a 300lb rider, on-road or off. That's due to what kind of loads they can structurally take. Whether the spokes are operating within their fatigue limit is a separate concern, and that's what determines whether you'll break spokes. It's also true that the margin for error in how well internal stresses have been relieved is going to be little if any here, possibly to an extent that exceeds practicality.