If the reducers won't hold, roto-zip have a straight 1/8 collet that might work, if you have someone that can ream a hole in a short piece of 3/8 or 1/4 round bar, ( it can be made from any material ) then you can Loctite the Burr in to the sleeve and than hold that with your normal collet
Here are 2 that should work also, make sure the Burr shank is not 3mm this will be loose in a 1/8 collet, if the Burrs you have are 3mm shank dia, then you will have to make the bushing as above to suit
https://drillbitsunlimited.com/14-to...-P4895344.aspx
1/4'' to 1/8'' Collet Reducer | Rockler Woodworking and Hardware