The tranny has to be taken down just about as far as it will go for synchros to be replaced, so it would be a little silly not to replace the clutch of all things. Really at that point you should make a list of components that possibly need addressed and heavily consider making the leap while you're at it.
And yes, ignoring your synchros can cause you to damage your gears if you're not paying much attention. Without them you'll find you'll have a TINY window to shift and will likely be grinding gears together until they're forced into place. Not to mention it sounds embarrassing, generally a 2/10 experience. There's a reason they've been common place for 60-70 years, and existed for a few decades before their acceptance; driving without them is pretty shitty
Edit: If our suspicions are correct and you don't want to replace it I'd recommend taking it to a highly rated transmission service shop near you for the work and not DIY in your garage. It'll be spendy (probably around $2k if I had to guess) but the headache it'll save alone makes it money well spent.