If you look close enough there will always be some carbide precipitation, even with L's. What you have to do is engineer your application and determine if that level of precipitation is acceptable. In almost all cases it will be through the use of the L grades.
Also, since any visual carbide precipitation criteria is arbitrary, or drvien by service requirements, you may want to do a re-evaluation of your rejection criteria and determine if indeed it is more restrictive than what your application rerquires. That would be my guess.
If you determine that the resulting precipitation is too excessive for your service, if your service requires the complete absence of any carbides (essentially impossible), I would suggest going to a higher alloy. Perhaps a nickel base alloy.

316L in an austenistic SS, so Waht is your consumible?

While it is not uncommon to experience grain growth in the weld and HAZ, especially HAZ, is is uncommon to see chromium carbide precipitation in those areas. This is known as sensitization, which is a function of time, temperature, and carbon content. Exceeding these will  lead to IGA, or sensitization. It is not the carides that lower the corrosion resistance, but the areas adjacent to the carbides that are detrimental to corrosion protection. If you have done a corrosion test and it passed, then evidentally it is so insignificant that it is acceptable. Also, the recommended interpass temperature for 316L SS welding is 150C, or 300F. Even a "L" grade SS can experience sensitization, but it needs to be at temperature longer than an "H" grade. Sometimes, during procedural qualification it is harder to maintain these temperatures due to the size of the test coupon. The longer it stays in the 800-1500F range, the greater possibility of IGA. Test coupons are not always indicitive of production welding, so a test coupon needs to have special precautions.