I think your needs would be served well by a Honda or Yamaha 2000 but it may be over kill for what you stated above. They are reliable and quite with the benefit of producing true sine current instead of modified sine waves which is nice if you are planning on using it to run sensitive electronics like TVs or computers. They are substantially more expensive then similar wattage "contractor" type gensets.
The typical onboard gensets that are installed in RVs are produced by OnanCummins are usually 4000 - 7500 watt units that produce true sine wave current. They are built to be installed in closed spaces with minimum air flow and have features like remote start (start stop from within the RV)
It may be an idea for you to consider a small contractor type genset instead of the Honda or Yamaha 2000s. They can usually be had for $150-$300 and would be powerful enough to run power tools (depending on the tool I suppose). Champion Generators have decent reviews and I know a lot of people that use them.
Depending on the space and available ventilation - it could be possible for you to mount the genset in a utility box in your flatbed and run an option remote start wire into your camper for added convenience. You would need a genset that has electric start instead of pull start only though for this to work.
