Depending on your budget you do have some options.  As you mentioned, control burning is a viable option for many species.  Pine trees are more fire tolerant than most other vegetation and in the grand scheme of things, burning is relatively inexpensive.  A regular burning program can help you get some of the problems under control but it might take a few years to achieve total control.

I'm not sure what resources are available in your area but aerial application of herbicides is also an option.  It is certainly more expensive than burning but it's more effective in the short term.  You can probably contact someone in your local department of natural resources/forestry that can give you a better idea of the cost.  

The bulldozer simply clearing the property would likely be the most expensive and possibly the least effective for long term control.  Clearing brush etc. can quickly remove the visible vegetation but the plants would probably sprout and grow more of what you're trying to get rid of.  

It's hard to recommend a plan of action without having seen your property (I'm in Mississippi) but I might lean toward prescribe burning and if necessary a follow up spot-spray herbicide application after two or three burns to help clear up any trouble areas.  I hope this is helpful in some way and you're welcome to email me if you have additional questions at msuforester69@gmail.com.