Newton did not measure forces for this purpose, nor could he, the second law itself is needed to make sense of measuring forces. What is actually measured are velocities and accelerations, or rather times and distances traveled, so any "measuring" of force has to presuppose at some point that the law holds (establishing the Hooke's law would have to presuppose it too, see Did Hooke's law come from experiments, or was it mathematically derived from Newtonian mechanics?). What Newton measured were the effects it predicted (for collisions, pendulums, circular motion, etc.), not the forces themselves:"On the same Laws and Corollaries depend those things which have been demonstrated concerning the times of the vibration of pendulums, and are confirmed by the daily experiments of pendulum clocks".
The notion of force in circulation before Newton was that of an "internal" force that imparts a body with a quantity of motion (our momentum) and keeps it moving, and can roughly be identified with $Fdt$ or its integral. For impulsive forces in collisions its action was instantaneous, and so imparted finite momentum in zero time. This is the notion that appears in Newton's early works. Eventually, he realized that this Aristotelian idea was incompatible with the Galileo's principle of inertia, and switched to calling the force that which affected a change in the state of uniform motion, $F$. "Tis known by the light of nature, that equal forces shall effect an equal change in equal bodies", he wrote, meaning $Fdt\sim dv$. If we denote the proportionality constant $m$ we get the second law $Fdt=mdv$.
"The light of nature" is Descartes's term for the innate knowledge of our reason. But the reason was helped, in this case, by the prior work of Galileo on the falling bodies and inertia, Descartes and Huygens on collisions, and Huygens on circular motion, which discredited the Aristotelian "internal forces". The OP quote is taken from Definition I of Principia, and refers not to establishing the proportionality of $F$ to $\frac{dv}{dt}$, but to determining that $m$, mass, is proportional to weight, what we call the equivalence of inertial and gravitational mass:
"The quantity of matter is the measure of the same, arising from its density and bulk conjunctly... It is this quantity that I mean hereafter everywhere under the name of body or mass. And the same is known by the weight of each body; for it is proportional to the weight, as I have found by experiments on pendulums, very accurately made, which shall be shewn hereafter."
The accurately made experiments are described in Principia, Book III, Proposition VI, Theorem VI:
"It has been, now for a long time, observed by others, that all sorts of heavy bodies (allowance being made for the inequality of retardation which they suffer from a small power of resistance in the air) descend to earth from equal heights in equal times; and that equality of times we may distinguish to a great accuracy, by the help of pendulums. I tried the thing in gold, silver, lead, glass, sand, common salt, wood, water, and wheat... The boxes hanging by equal threads of 11 feet made a couple of pendulums perfectly equal in weight and figure, and equally receiving the resistance of the air. And, placing the one by the other, I observed them to play together forward and backward, for a long time, with equal vibrations."
As expected, the conclusion presupposes the second law, and the third one as well. To verify the third law, Newton collided two pendulums with unequal masses, as described in the Scholium to Axioms, or Laws of Motion in Principia. The impacts/forces were "measured" by how far the pendulums rebounded.
Fowler gives a brief account of Newton's evolution on the notion of force, and the second law, in Newton Clarifies the Concept of Force lecture, for a more detailed account see Westfall's biography of Newton, Never at Rest.
