I suppose it's not impossible that could happen if compressor and dryer motor started absolutely simultaneously but, despite echoes' experience, I would have thought it extremely unlikely.
It takes at least around 22A to blow a 13A plug/FCU fuse ever, and a lot more current than that that to blow the fuse very quickly (i.e. if the excessive current is very brief, as with a start-up current). The running current of your fridge is only a bit over 0.3A, so even if the start-up current of the compression were (for a second or two) 20 times that (which I imagine is unlikely), that would not be much over 6A or so.
The dryer's start-up current is more difficult to estimate, since the great majority of the 2600W is the heater, rather than the motor. However, I would have thought that if dryer+fridge was likely to blow a fuse if they both 'started up' simultaneously, it would not be uncommon for a dryer alone to do it - and that's not something we hear of happening.
If it were me, I wouldn't be at all concerned, but it's obviously your choice/decision. Looked at pragmatically, if I'm wrong not to worry, it would only happen once (since I/you would then realise the need to change the electrical supply arrangements) - so, although annoying, would not be the end of the world.
Kind Regards, John