Not being able to break into the manifold makes it tricky. Can you send a circuit? Here's a few suggestions anyway, hopefully one of them has some potential.
If the actuator always works at the same sort of pressure (when running properly) then you could consider teeing a small, rapid response relief valve into the LS line from just this actuator. That way, when you started up, the full magnitude of the high pressure pulse wouldn't make it through to the pump controller and you wouldn't be stuck waiting for the pump pressure to back off because it never went high in the first place. [Don't forget to put an orifice upstream of the tee point otherwise the relief valve will be swamped and won't work the way you intended.]
A soft switching directional control valve might help, you haven't said what sort of on/off control you have. Some cartridge valves are readily interchangeable with their soft switching counterparts. The soft switching period is only a matter of tenths of a second but it can just take the sting out of the on/off pressure pulse.
You could fit a new [pressure compensated] flow control valve directly on the actuator port and create a new LS line from the actuator port itself (unless the manifold is mounted straight onto the back of the actuator). Abandon the original LS line and fully open the original flow control valve.
I'm pretty sure you will have done this one already, but, do make sure your LS lines are bled of all air - including any dead ends, and make sure the pipe routes don't allow the LS line to empty under gravity when the machine is shut down. If you have hoses in your LS lines then try to replace them with rigid tubes, smaller/stiffer hoses, or fit check valves so the bare minimum of volume suffers the high pressure pulse [this minimises the accumulator effect of all your LS line volumes].
If your pump (what make by the way?) does have a bleed off orifice then try changing it for one with a bigger hole. Or introduce a new bleed off orifice, maybe on a small solenoid valve so that the extra orifice was out of circuit for all other users.
Check that the LS compensator spool in the pump controller isn't gummed up with grease or sticky stuff, and that it is moving freely in its housing. It wouldn't be the first time a pump controller had the wrong spool in it. Some LS compensator spools differ from the pressure compensator spools only by virtue of an orifice down the middle of the spool - check they haven't got mixed up at some point. You could even ask the supplier to send you a spare compensator assembly off an old pump just to see if that eliminates the problem. [The compensator wears much less than the pump pistons and bearings - a compensator from an old pump might just be run-in enough to make all the difference.]
Kind Regards
DOL