Electronic – Microcontroller with >30KB RAM and high performance in a TQFP-44 or similar package

In principle this is good candidate for FPGA based design. Regarding your requirements:

ad 1. The FPGA most likely will be more expensive, by how much that depends on the device you choose. At first glance smallest Spartan 3 from Xilinx (XC3S50AN) will be more then enough for this task (~10£ from Farnell). I think you can assume this is upper boundary for the cost (it has 56kB RAM inside, so it is more then you need). You may find cheaper device either from Xilinx offering or their competitors Altera and Lattice.

ad 2. The package is the tough issue, I did not saw FPGA with smaller footprint either. However maybe you can use CPLD device (for sake of argument the CPLDs are small FPGAs) which may be in smaller package (PLCC or QFN). On plus side they will be cheaper (even single $) on negative side most likely will not have RAM inside. With CPLD probably you would need external SRAM chip.

ad 3. FPGAs and CPLD current consumption is highly dependent on the programmed design. However there is good chance that FPGA and especially CPLD design would consume less than your current solution.

ad 4. FPGA do have that kind of memory inside, CPLD most certainly not. This may be solved by external sram chip (or two). For example:

|SRAM 1| <--> |CPLD| <--> |uC|
|SRAM 2| <-->

In such arrangement while the uC is writing to SRAM 1, the CPLD will be displaying data from SRAM 2. The CPLD should be able to handle both task simultaneously.

Of course you can solve this in other ways too:
1) use faster uController (ARM for example)
2) use device with some programmable fabric and uC inside (for example FPSLIC from Atmel, however I have never used such devices and I know very little about those)

Standard disclaimer -> as designs are open problems, with many constrains and possible solutions whatever I wrote above may not be true for your case. I believe it is worth checking those option, though.

Okay, so the TL4242 is too expensive. No problem, you can still make a current source using more common components.

The current is defined by the input voltage as \$ I_{LOAD} = \frac{V_{IN}}{R_{SENSE}} \$. Choose \$R_{SENSE}\$ in function of the microcontroller's \$V_{DD}\$. If you supply \$V_{IN}\$ with the PWM signal the current will switch between \$I_{LOAD}\$ and zero.

edit
I hadn't looked at the price for the TL4242, but Federico is right: the price is very reasonable. You would spend more if you want to PWM control the LM317.

edit: driving 2A
One thing to keep in mind when selecting components is that the transistor replaces your series resistor which usually controls the LED current.
If the transistor were to be purely a switch you would want it to be as efficient as possible, i.e. dissipate very little power. In that case a MOSFET would be the best option, there are lots of them which have \$R_{DS(ON)}\$ of less than \$100m\Omega\$.
Now that the switching element is a current limiting device this is not that important. You can use a BJT here. Power transistors for 2A \$I_{C}\$ don't have high \$H_{FE}\$, which means that the opamp would need to supply a rather high current into the base, and this may be beyond the opamp's capabilities. A Darlington transistor like the TIP110 is the solution.