$4K is steep, but unless you can find a full carrier or a hotshot with partial loads that are in the same neighborhood on both ends, you are going to be hard pressed to get below $3K for a run.
Think about it:
Fuel - $800 (10 mpg fuel + DEF)
Time - $800 (4 days at $200)
Tolls - ?
Equipment Depreciation - $200 ($0.10 per mile)
Maintenance (tires, PM service, incidental repairs) - $160 ($0.08 per mile)
$1960, bare bones minimum cost for dedicated run.
If you add to another delivery the costs can be split with other customer, but there will also be added costs of multiple pickup and deliveries. The last truck I booked to run 1600 miles with no tolls and minimal low truck speed limits was $3300.
If you are talking one of the three west coast states and/or New England, drivers will charge for the pain and nuisance of dealing with regulations and costs.
If you find anyone for less than $3K for this run, verify everything! DOT, MC, insurance, and carrier ratings.