Definitely Read the Flipping Manual!
Very clear this is an Industrial machine designed to run on an industrial type supply, not an ordinary 13A domestic socket. The 16A plug is surely a strong hint!
Unfortunately, the manual doesn't specify input current or power. However the 16A plug means a 13A spur will supply the cutter, even though overloaded. So far so good.
Unfortunately, putting a 13A plug on the machine breaks all the requirements listed in manual. The warranty is void, the cutter no longer meets even slack industrial EMC requirements and it's highly likely to annoy neighbours. If the house burns down it's 100% the owners fault. Best hope the insurer doesn't find out, because most policies don't cover damage caused by operating industrial machinery and especially not industrial machines electrically modified by an over-confident hobbyist!
Safer ground if the machine is plugged into a professionally installed 16A socket because it's more likely the supply meets the necessary standard. Otherwise fitting a 13A plug means the owner is taking a risk.
I'm all for risk managed engineering, but it does require an adequate understanding of what might go wrong, what the impact will be, ways of mitigating the risk, and a recovery plan should the worst happen.
One risk is poor EMC wiping out radio signals – radio, tv, mobile phones, satellite, and WiFi. Impact ranges from zero, because they don't find who's causing it, to getting duffed up by an angry football fan after you zapped the cup final climax. Expect policemen if an essential radio service is disrupted, followed by fines and confiscation of equipment. None of this is likely is the cutter is operated intermittently in short bursts and there are no close neighbours: it's just anti-social.
More serious is the prospect of damaging cables inside the walls by overheating the insulation, perhaps slowly baking it until it disintegrates and causes a house fire. Fitting a 13A fuse only partly protects against this because fuses can actually take 20A continuously, 50A for 100 seconds and 80A for 1 second. If I owned one of these cutters I'd measure how much mains current it actually draws when cutting: below 13A, I'm happy (apart from the EMC). Anything in the 15-40A range is unacceptably risky because a 13A fuse allows the cutter to work for a long time in short bursts, with dodgy consequences.
Dave