All sine to triangle conversion schemes are doomed to expense, complexity and failure.
I think you've made a trap for yourself. Stay out of the rabbit hole!\aq
Filter+AGC could work but it will be complex to implement. Log amps are monopolar and have horrible temperature coefficients that are hard to compensate.
Analog multipliers are a lot less bother and relatively inexpensive, eg AD835. There are voltage and digitally controlled amplifiers: VCA's PGA's an VGA's, that perform two quadrant multiplication. The "loop filter" in an AGC circuit impacts the response time and the AGC will struggle to operate at low frequencies.
I suppose you could use a PLL to multiply the squarewave frequency by a binary value 'n' and then clock the output of an n -bit an up/down counter. Connect the output to an ADC? Possible. An arduino could possibly be forced to perform the frequency multiplication and output a 10 bit resolution triangle wave.
https://bb7.com/e-unum-pluribus-using-arduino-as-a-frequency-multiplier/ Mr Hess is right. Creating a very good quality sine from a triangle is far easier. Many classic precision lab function generators did just that.
https://till.com/articles/sineshaper/index.html See also
https://lookmumnocomputer.discourse.group/t/vco-3340-sine-wave/1337.
A classic analog Voltage Controlled Oscillator (VCO) makes this simple. Example attached.
The integrator section can produce both negative and positive slopes from the control voltage.
The sign of integration is switched by the MOSFET. When the integrator output trips the comparator the slope is reversed and a new cycle starts.
This circuit has an output frequency that is linearly proportional to the control voltage. About 1kHz per volt with the values shown. Regulated power supplies are needed for frequency stability.Its easy to rescale the frequency by changing the comparator resistors and the capacitor. R1=R3=R4, R2=2R1. Note that not all opamp types can be used as comparators.
Add gain/ Attenuation, summing amps etc to set output levels.
Instrumentation amplifiers are ideal for these tasks as they give you sum/ difference and slope. (mx+c operations)
For a total solution DDS is the low cost and simple way to go. eg AD9837