How about a serial DAC like MCP4921, followed by a power op-amp, or tweaking the feedback pin of a linear or switching regulator?
1.5A may be a bit above the range of most LED/backlight drivers, but that comes to mind as a product line that's tailored around supplying constant current. Some are continuously dimmable (rather than PWM gated), or you can do the "tease the feedback" trick all the same.
Also consider normal regulators, the ones with current mode operation, and a transconductance error amp, with external compensation. These, the COMP pin voltage corresponds to output current, so just overdrive that directly. Mind, there's usually peak detection and slope compensation involved, which mean the setpoint will not correspond to current exactly, and you may want an old fashioned current sense resistor (maybe with a sense amp) and error amp to get an accurate response.
Or you can use a regular regulator in a low-side-sense current feedback configuration, which drops VREF across the sense resistor. Which sounds annoying, but bear with me. If VREF is 1.24V say, that's kind of a lot (several watts at 1.5A), but you could use an amplifier to boost the signal, so that only say 124mV is needed (10x amp). Preferably, you can find a regulator with a lower VREF; 0.8V is pretty common (0.8V * 1.5A = 1.2W). Not sure if you can find lower.
Tim