Q: "In Newton's time, were observations accurate enough to corroborate that the Sun orbits the common center of mass of Sun-Jupiter?"
If the question means to ask for observational corroboration of a directly-observable difference in center of motion, then I believe the answer is that nobody in the 17th-century could do that. While many of the distances between solar-system bodies were known approximately on a relative scale -- effectively by triangulation -- all the measures were of limited precision, and a few (such as the earth-sun distance) were very poorly determined indeed and very difficult to measure with any improvement (triangulation was not available, and attempted measurement by parallax was then very rough at best, see Albert van Helden's 'Measuring the Universe' (Chicago, 1985)).
On the other hand, all of Newton's conclusions about the solar system in Book 3 of the Principia were heavily dependent on observations. The major types of observations on which he relied were set out at the beginning of Book 3 in the section headed (in the 3rd edition) 'Phenomena'. The logic connecting these observation-types to many of his conclusions necessarily had to be indirect, and the chain contains multiple links, but the logic is there, and so the observations on which he relied do indeed -- in that indirect sense -- corroborate his conclusions about the motion.
One of the things that Newton showed in Book 3 of the Principia, by a series of arguments concluding with Proposition 12, was that
"The sun is engaged in continual motion but never recedes far from the common center of gravity of all the planets."
(He had already moved beyond considering only Jupiter.)
(This language of Prop.12 is quoted from the 1999 Cohen-Whitman English translation from the original Latin. The 1999 translation is not generally available online -- unfortunately -- but the earlier (1729) English translation is online, and it is still very useful as long as one has some patience for the antique style and language. It can be found here -- the link points to p.232 where Prop.12 of Book 3 begins.)
Proposition 12 doesn't stand by itself, of course, so one has to follow the preceding sections where the arguments for relevant intermediate conclusions are presented. These include assessing the (relative) masses of planets with satellites -- as mentioned in the question -- by the technique based on the satellite's orbital period and the size of its semi-axis. This is Newton's own method, described at Corollary 1 of Proposition 8 (beginning page 227 in the 1729 online translation).
Several kinds of care are needed for differences in the language of 300 years ago (and also for the scattered locations -- in Newton's book -- of the different pieces of his arguments). One point of care that is relevant to the sun's motion about the solar system center of gravity arises from "Hypothesis 1" (located just before Propositions 11 and 12 in Book 3): "The center of the system of the world is at rest."
(a) 'Hypothesis' in Newton's early writing had its old classical meaning, i.e. to denote the starting-point(s) for some discussion or argument, or an underlying e.g. necessary supposition. It was only later -- after one of Newton's heavily critical reviewers of the first edition of 'Principia' took the word 'hypothesis' in a quite different sense, 'unsupported conjecture' or even 'baseless fancy' -- that Newton made his well-known negative remarks about hypotheses (in that latter sense).
In the first edition of Principia, all of the phenomena at the start of Book 3 had been labeled 'Hypotheses'. But these were far from baseless conjectures. Newton had taken care to select and rely only on phenomena that were agreed on by the main astronomers of his day, thus, about as well-established and sure as they could be. By the third edition, Newton had made a clarification, using for them instead the designation 'Phenomena' -- most likely, to avoid further instances of the misunderstandings and disputes over the word 'hypothesis' with which he had been made familiar in the meantime.
(b) {center of system of the world at rest}: In the 3rd edition Newton retained 'Hypothesis' for the supposition he would adopt about the center of the 'system of the world' -- that it is at rest. This was a hypothesis in Newton's old sense, a starting-point or supposition for the discussion to follow. 'System of the world' effectively means what we now call the solar system. Newton clearly wished to address the point that was controversial in his time, the question whether the system is geocentric or heliocentric, and as described below his conclusion was in effect 'nearly heliocentric but not exactly'. His arguments call in aid several points:
-- that by the first law of motion and the corollaries to the laws of motion (especially cor.4) it is indifferent whether a system of bodies is on the whole at rest or in a state of direct uniform motion; and if a common velocity or acceleration affects all the bodies in the system equally, then there is no perceptible effect on their mutual relative configurations or motions. Newton's 'Hypothesis' adopts the common/popular position that the center of the world, whichever it is, is at rest, and he is content with that choice as a 'hypothesis' in the old sense, arbitrary though it may be, because he clearly knows that the choice either way leaves the rest of his arguments unaffected.
-- that in a system of two (or several) bodies in mutual motion, their center of gravity is either at rest or in direct uniform motion; and
-- that the relative mass of the solar-system bodies with satellites (relative to the sun's mass) could be estimated by the method he has described in Prop.8 Cor.1.
With these dynamical arguments founded ultimately on observations, Newton arrived in Proposition 12 at remarkably modern conclusions. The system is very far from geocentric (this is mainly from the relative smallness of the earth's mass compared with the sun and the others), but while it is nearly heliocentric it is also not exactly so, its center is the center of gravity of all the bodies involved, and the sun, while close to that center of gravity, is not exactly on it:
... [even] if the earth and all the planets were to lie on one side of the sun, the distance of the common center of gravity of them all from the center of the sun would scarcely be a whole diameter of the sun. In other cases the distance between those two centers is always less.
The way in which that conclusion stands up today is neatly illustrated by diagrams that have been produced for the relation over time between the sun's center and the solar system barycenter (center of gravity), see e.g. here.
