Improving a simple bipolar current limiter
If you don't want Vbe dependence, don't use B-E junctions for voltage references.

One way to reduce the Vbe dependence is to use a Zener diode for most of the reference, so the added Vbe junction voltage will introduce a relatively smaller error.  For example:



The reference voltage in this case is the zener voltage minus the B-E junction.  If the B-E drop is 600 mV, then there will be 5 V accross R1, which allows for 4.2 mA to flow.  Adjust the values as needed.  R2 provides a minimum guaranteed bias current for the zener.

One drawback of this method is that the zener voltage eats into the compliance range.  Everything is a tradeoff.  If you want more precision, then using a opamp driving the gate of a FET with current sense feedback resistor is a better option.  There are lots of current source topologies.  What is best depends a lot on what performance you need, and space and money you are willing to pay for.  You haven't said anything about the accuracy and compliance range, so any concrete suggestions are impossible.