I think you might be looking at the wrong problem here, i would suggest you check the nuckles on the rear of the gearbox and the remote/gearchange selector toggle bolts for tightness & correct alighnment and the 1st/2nd selector fork inside the gearbox for wear & torque of the retaining bolt. Yes a tight idler gear will cause all the gears to have a selection problem, and a loose gear will rattle but you only have the problem of 1st gear being hard to select. Even check the sliding hub to make sure its ok and that is has the correct springs and they are not broken.
In my experence the gear selector parts would be the first thing to check before assuming theres a fault with the idler/primary gear.

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I have got my car back on the road and the gearbox is now smooth as!
Easy into 1st gear and no "snick" when changing up or down thru the gears.
Very happy.
Just annoyed that some one decided to make dodgy gaskets?

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1969 Cooper "S"
1967 Mini Deluxe
1973 Clubman Van (the fleet spare)
1978 ex 1275 LS ("Wizard" Eaton Supercharged)
2015 HSV GenF GTS(occasional drive & tow car)
2019 MINI F55 Cooper S

thanks - points noted

the box was perfectamundo before changing the gasket in March - nothing else changed...

It's hard to say whether it was tight getting into the other gears, because you're selecting them on the move. When changing back to 1st while doing motorkhanas etc. there's no selection problem because I've got the full weight of my arm, it's a different action to gently sliding it into 1st gear at the traffic lights

I'd rebuilt the box 12 months earlier, selector forks were fine, back then. I checked the alignment of the knuckles etc. when I pulled it apart a couple of weeks ago. The only thing amiss was the idler gear, and it has been lunching on the transfer casing.

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did I tell you that I won a trophy?

Yes My issue was exactly the same.
Gearbox had most parts replaced including selectors and gears and was great before the last engine rebuild and the use of a "skinny black" gasket for the Clutch housing.
Now I have used the correct orange gasket and double checked the idler gear measurements all is good. Except the new paint burnt off the extractors

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1969 Cooper "S"
1967 Mini Deluxe
1973 Clubman Van (the fleet spare)
1978 ex 1275 LS ("Wizard" Eaton Supercharged)
2015 HSV GenF GTS(occasional drive & tow car)
2019 MINI F55 Cooper S

we won't do that again, will we!

looks like I'll be stuck with a skinny black...

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did I tell you that I won a trophy?

You need to check the idler gear clearence BEFORE you bolt the gearbox to the engine, there are different thickness gaskets around and they have been a problem for years. 003 to 006 is the clearence you need, bolt the transfer housing to the gearbox with the idler gear & shims fitted & measure the clearence with feeler gauges. Its allways nice to have lots of shims on hand to do it properly but if you only have a couple to choose from then you can use the different size gaskets to you advantage, you can linish or surface grind the shims and if your desperate you can mount the transfer housing in a lathe & remove some metal from the gasket surface.

_________________
Have a Nice Day.
If already having a nice day please disregard.

Been rebuilding mini engines since 1968 and never had this problem before...

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1969 Cooper "S"
1967 Mini Deluxe
1973 Clubman Van (the fleet spare)
1978 ex 1275 LS ("Wizard" Eaton Supercharged)
2015 HSV GenF GTS(occasional drive & tow car)
2019 MINI F55 Cooper S

that's the plan... like I said, I'd set it up properly last year, but neglected to check it in March when I was in a hurry - always happens doesn't it

I was going to sit the transfer case under the mill - I've got a few transfer cases, but the bearings in it are worth more than the time it'll take me to trim this one - hopefully I won't need to

I've got a good collection of thrust washers

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did I tell you that I won a trophy?