Large current spike is a .usual thing for most power supplies.
Only if they are designed by an idiot.
@Lango1818: I've no solution except: stop being a cheapskate. If you want a lab supply, get one and pay what it costs. Anything else is landfill waste.
All CV/CC lab supplies will have this issue, no matter how expensive. The problem is caused by the need for very low - ideally zero - output impedance in CV mode, which is the normal mode of operation. An output capacitor is always used. This works along with (and part of) the CV control loop to provide a very low effective output impedance.
For CC mode, we want ideally infinite output impedance. This implies zero output capacitance. Of course, neither infinite nor zero are realizable. Unfortunately, due to the presence of the output capacitor, the CC mode of typical lab supplies is very much impaired. The supply can only truly regulate the
average current, not the
peak, because the output capacitor is not within the CC control loop, and it can supply large peaks.
This problem is often worse with switch mode power supplies than with linear ones. The switchers will usually have more output capacitance as a consequence of their design. In a linear power supply, most of the bulk capacitance is prior to the regulator, and the output capacitor can be designed to be quite small, and often specifically has some high-ish ESR (up to a few Ohms), since very low ESR can disturb the regulation loop. On the switcher, the capacitance will often be 100s to 1000s of uF, and always low ESR. The best switch mode lab supplies follow the switcher with a linear regulator. This linear regulator can and will have a smaller output capacitor, but it will be be present.
Some true CC lab supplies exist, e.g. HP 6186C. These are generally a true CC source with very low (10s to 100s of pF) output capacitance, combined with a shunt regulator for voltage limiting. A supply designed this way is always sourcing the programmed current. If the load is too high resistance (or disconnected) then some or all of the current will be shunted by the CV voltage compliance limit portion. This makes them very inefficient, and they actually run hottest when no load is connected. But you can do things like set it to 5 mA @ 100 V, then connect the output across an unknown zener diode and measure the breakdown voltage of it. You can't do that with a typical lab supply, because the output capacitor charged to 100 V will dump into the zener, destroying it instantly/violently.