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..... a small router , try a laminate trimmer . Either one will take the router bit you need to clean out the existing groove . I've always assumed that's how they made the original groove in the 1st place .
You'll need a flute bit of the correct size to match the spline you intend to use . Splines come in various sizes , match the bit to the spline and the cane gauge you'll use .
Cut the existing grove a tad larger if you want ... or cut it exact , or cut it a tad smaller and hand clean what's left ... I'd go for the tad larger personally . There's no constraints , only the spline match the groove and cane gauge .
Glue ?? I don't know . You don't need it for anything I can think of if the spline is the correct size for the groove ... the spline isn't going to come out in my mind , moisture only makes it tighter in there . But glue can't hurt anything I guess .
Varnish for exterior use cane ... very thin (cut coat) on top only if that's what you want , but I wouldn't , natural I like . Can't hurt anything to set the spline in a varnished groove though .
I've heard of oil stain on cane , but not done that myself . If oil stain , I'd test a scrap piece of the cane mesh 1st to see how well it absorbs and if I liked the way it looked . I'd think a penetrating stain , not surface type , but not so sure it would penetrate well either , cane is hard and slick .
Use the router guide or make a jig guide . Some freehand may have to be done on intersections where a radius already exist . Freehanding a small laminate trimmer is pretty easy for a short radius . Have same thickness support stock tight up to item you're grooving if needed for router base support .
I cut hinge gains freehand all the time with a laminate trimmer , only using the trisquare as a guide at the very ends , all edges razor scored on layout before routing out . At one time (before pre-cut gains) in commercial production new door hanging , the full door hinge jig is best to use , no layout after intial setup , no misses , quick repetitious hinge gain cutting production . Sure beats the old hammer and chisel which how I had to do it for many years in the beginnig . .
Next time try the little laminate trimmer as your router (buy one if you don't have one , you won't regret it , does all other smaller scale routing work too) ... use the attachable guide with it for paralell longer runs , same as you'd use an attachable guide with a circular saw .
Forget that digging , witling and scraping with whatever tool works in the groove ... route it out a new !!