Hi Colin, your question doesn't have a simple and straightforward answer, but here goes.
The short answer is yes, the 10 amp switch will work OK, it'll stand the use for a time period which might be quite difficult to predict, but it's unlikely to go snap crackle pop immediately, so give it a try and see how you get on.
BUT>>>> (number 1)
Switch gear for controlling motors is rated to allow for the inrush current. When you start a motor, it accelerates from stand still very quickly, and while it is doing so it will take (if the supply will provide it) typically 6 – 9 times the "rated (full load) current". The rated current is the current while providing the full load torque relating to the power of the motor. Power equals torque times speed, so the rated current shown on the motor rating plate assumes that the motor has already accelerated up to the speed at which it runs at the corresponding load torque, But the switchgear has got to survive the shock of getting the motor onto its load curve.
This inrush current is brief, but severe. Most drum or cam switches are designed for switching a motor, so they have a rating for what is called AC3 duty – this is a number that allows for some de-rating of the switch. The start duty of using it upstream of a motor is only a brief torture, most of its life the switch will only have to carry no more than the rated motor current. But while it is carrying that start inrush current it has a hard life, and the "rated current" quoted for the switch has to allow for this. There is also something called AC1 duty – this is closer to the simple use of the switch to control a resistive (e.g. heating) load and will be a higher number that the AC3 figure specified. So which duty is the "15 amps Dewhurst switch"? Well, it is almost certainly an AC3 figure, because a Dewhurst switch's reason for being is to control a motor, but when you buy a 10 amp switch you need to be clear what the 10 amp figure actually represents. If it's a 10 amp push button switch, and the 10 amps is an AC1 duty then it is probably (guessing) only a 6 amp AC3 motor duty rated switch and the magic smoke will escape if you connect it to your 8.7 amp rated motor.
There is a complication here that the concept of AC3 only applies to 3 phase switchgear, and single phase motors are a quite different beast to start, but that's too complicated for this time of night!
Now, let's develop this a bit further. You have mentioned a push button type switch – and this also affects the answer. If your plan is to fit a forward/reverse selector switch AND a push button ON/OFF switch we're into a different can of worms. If the drum/cam forward/reverse selector switch is only ever going to connect the motor wires together without them being energised, and the switch of the inrush (starting current) is done elsewhere then you can over-rate the selector switch significantly. Nobody's going to commit to a definitive statement how greedy you can be, but your 10 amp switch controlling a 8.7 amp motor will be fine. The motor rated current is below the rated current of the selector switch, so all is well. The push button starter now has to make the motor inrush current, and break the motor voltage to stop it again, while the selector switch just has to carry the motor load current. It's the making and breaking reliably which is the arduous duty, by comparison any contacts carrying the motor current which don't have to make or break the circuit and carry current at the same time have an easier life.
However I'm bound to muse on "BUT" number two. My experience is that it is unlikely to be the switchgear that causes an RCD to trip, it is much more likely to be a fault within the motor. The current and voltage figures you quote say this is a single phase motor, and that rings alarm bells straight away. Single phase motors have a capacitor that'll happily go fault to earth, and if that's not the problem the centrifugal switch inside the motor which controls the start winding (if there is one) has a REALLY hard life, and splatters conducting metal vapour over its insulating mounting plate. So what leads you to believe that the drum switch is the faulty component? Has the switchgear been re-wired recently – and if so has the problem only arrived since something was changed? Have you had the insulation of the motor checked – it's a very easy and quick measurement for someone to do with an insulation tester that could stop you spending money on new stuff which doesn't fix the problem. My suggestion is to take it to a rewinder and get them to check the motor over before spending money on switchery. But maybe you already tried that.
Yada yada yada. Like I said, it's a simple enough question, but the answer is anything but!
But I sympathise with the aggro from SWMBO – I had to feed my shed with a separate RCD circuit for all the same reasons, though that was caused by the leakage currents of VSD's.
Hope this helps
Simon