Step-down with external mosfet - why heavy current surges?
I'm amazed it works at all.  

You've completely omitted the single most important component of any buck converter, of course it isn't working.  

Where are your input capacitors?!

Your voltage source (presumably a lab power supply since you mention triggering it's current limiting) should not be even care about the current spikes because all the current should be coming from your beefy input capacitors which are placed as close as possible to the anode of the switching diode and source of the P-channel fet.

The yellow is the input of the mosfet? You should not be able to tell when the MOSFET is on or off by measuring the input to it, because your input capacitors will maintain the voltage.  You're powering this directly from your lab power supply without any input capacitance at all, aren't you?

Buck regulators work by drawing large but brief amounts of input current.  You need to understand - MOSFETs give dV/dT rise and fall times the likes of which the ancient, slow, and terrible LM2576 doesn't even know were possible.  You need some solid ceramic capacitors millimeters (like, single digit numbers of them) from the MOSFET input and the diode anode.  You also need some nice and chubby electrolytics near the ceramics to provide the bulk energy storage you need.  The input ripple on the MOSFET should be below 80mV.  More than this, and you'll probably exceed the ripple current rating of any electrolytic capacitor on the input regardless of how large - even at 52kHz, many electrolytic capacitors will be essentially resistive and will just get hot from the ripple.  Reduce it to 80mV with ceramics however and they should be safe. Your ripple looks to be...well, it's definitely not below 80mV, let's leave it at that.

Also, that regulator is internally compensated so it is probably the worst choice for what you want to do.  Even if correctly decoupling the input of your circuit solves the first problem, I would not be that hopeful of it working with your instrumentation amp thrown in.  An entire slow-as-a-diddly instrumentation amplifier crammed into your feedback loop is going to crush what little phase margin the LM2576 had left and it will just be a glorified chaotic oscillator.  And you won't be able to do anything about it because the output of the error amplifier isn't broken out to the frequency compensation pin so you cannot do anything to stabilize.

Also, that regulator is old, expensive, slow, terrible, and the only reason you're using it is because you found that same old circuit online.  It's not a very good circuit, especially for your application, I would recommend getting a real buck controller chip that wasn't made when most people had dialup internet.  Seriously, if the LM2576 were a person and lived in Ohio, it's old enough to vote. 

You're already using a discreet pass element - the PMOS transistor.  Only, you're using it with something never designed to be used in this way.  So you're literally doing what you would have to do with a proper controller anyway, only making it much much harder on yourself.  Plus, modern controllers have the features you want - constant current and constant voltage regulation - built in.  There are usually schematics you can outright copy unchanged for your application.  This puppy might be a simpler chip to look into, off the top of my head.  

Also, a photo of your physical circuit would be helpful.  That is the most important bit of information when dealing with switchers.  Far more important than the schematic.  Both are optimal of course, but if you could only give us the schematic, or the layout, the layout is the better choice.