This is a real stumper! When I use charcoal, I lift the smoker off of the pan (see pic above), then start the coals. Once they're ready to go, I lower the unit back over the coals. With the lid on, without a water pan, if the fire is sufficiently hot, it takes less than two minutes to get up to 225, which is where I usually cook.
Are you using a water pan?
Are you igniting your coals externally, then adding them when they're good and hot?
In my experience, a full pan of lit coals will get me way over 225. I use more coals than in the pic above, but not much. Adding wood on top of that will produce even more heat, though it doesn't last as long. I've learned through trial and error that it's easier to start small and add one or two unlit lumps than it is to pull out burning ones.
BTW, most people seem to think that soaking wood isn't really necessary, especially if you're using larger chunks.
If you're still getting to know your smoker, why not throw a cheap meatloaf on there and see how it goes? That way you won't feel bad if you have to toss it, but you can still enjoy it if it turns out well.
I hope we can get down to the bottom of this, because smoking's a whole lot of fun, and it can definitely be done on these cheap Brinkmann units.