Noise at peak of triangle wave
Looks like you've run into a limitation of low power opamps.

This article mentions the effect you are seeing, and says that it has to do with the inability of the opamp to deliver the current needed at the peaks of the triangle wave.

From the article, I see three ways you can reduce or fix the glitches:


Use the circuit suggested in the article.  That has a schmitt trigger helping the opamp.  That's out of the question since you are only allowed to use opamps.
Use larger resistors and smaller capacitors to get the same frequency at lower loading of the opamp.  This has the disadvantage of being noisier and more susceptible to picking interference from other noise sources.
Use an opamp that is less "low power."


Point (2) is the easiest to try out.  If you have a selection of different models of opamps, then you may find one that you can plop in to take the place of your LM6171.

I expect the real solution will be a combination of (2) and (3) and a restriction on the maximum frequency you can use the circuit at.  That is, you make your circuit and specify "can be used at frequencies up to \$F\$ with no more than X% distortion."