Before European influence circa 1600, did any Chinese believe the Earth was spherical, and did they ever try to measure it?

Yes, some believed in a Round Earth, but we don't know if they tried to measure it.

I'm going to start from the basis of the Round Earth concept; Hellenistic Astronomy was based upon Babylonian astronomy (which was more than likely taken from Sumerian astronomy, the earliest civilization, so far as we know). Greek astronomers like Pythagoras, Aristotle, and Eratosthenes developed their theories based upon the observed science that came before them through Babylon, and much of what we attribute to Hellenism should actually be owed to the Babylonian and Sumerian civilizations. These Mesopotamian civilizations did not believe in a flat Earth concept. (the apple fell far from the tree). see: Babylonian Astronomy

In Babylonian cosmology, the Earth and the heavens were depicted as a "spatial whole, even one of round shape" with references to "the circumference of heaven and earth" and "the totality of heaven and earth". Their worldview was not exactly geocentric either. The idea of geocentrism, where the center of the Earth is the exact center of the universe, did not yet exist in Babylonian cosmology, but was established later by the Greek philosopher Aristotle's On the Heavens. In contrast, Babylonian cosmology suggested that the cosmos revolved around circularly with the heavens and the earth being equal and joined as a whole. The Babylonians and their predecessors, the Sumerians, also believed in a plurality of heavens and earths. This idea dates back to Sumerian incantations of the 2nd millennium BC, which refers to there being seven heavens and seven earths, linked possibly chronologically to the creation by seven generations of gods.

With that being stated, it's not impossible to believe that these views spread in some form to the Indus Valley and from there to the Shang Dynasty (cultural diffusion), given the span of a few thousand years. Therefore, the Round Earth view more than likely existed, not as the dominant ideology, but as an emergent one under scrutiny that was absorbed. Perhaps some ancient Chinese astronomers believed in a Round Earth, but like the Greeks, that knowledge was lost and a flat Earth concept became the dominant view through syncretism.

What we do know is that Yu Xi was the first Chinese astronomer documented to challenge the flat Earth concept in China. He held the basis of a heliocentric model but did not expand upon that, since he had to challenge the concept of a spherical Earth and spherical bodies in the heavens.

Extract from Yu Xi's wiki:

Yu Xi wrote a critical analysis of the huntian (渾天) theory of the celestial sphere, arguing that the heavens surrounding the earth were infinite and motionless. He advanced the idea that the shape of the earth was either square or round, but that it had to correspond to the shape of the heavens enveloping it. The huntian theory, as mentioned by Luoxia Hong (fl. 140-104 BC) and fully described by the Eastern-Han scholar-official Zhang Heng (78-139 AD), insisted that the heavens were spherical and that the earth was like an egg yolk at its center. Yu Xi's ideas about the infinity of outer space seem to echo Zhang's ideas of endless space even beyond the celestial sphere. Although mainstream Chinese science before European influence in the 17th century surmised that the Earth was flat and square-shaped, some scholars, such as Yuan-era mathematician Li Ye (1192-1279 AD), proposed the idea that it was spherical like the heavens.

This expands upon what was shared in the other answer. Some ancient Chinese scholars believed in a spherical Earth despite the ideas not being popular to the mainstream beliefs of their time.

I am not sure about 1600, but there is a famous book on astronomy called Zhou bi suan jing, which is usually translated as "The Arithmetic Classic of the Gnomon and the Circular Paths of Heaven". It is not known when precisely it was written, but it was commented by Chinese authors in 3d, 6th and 7th century AD. The book is full of problems like this: Suppose that the Sun is in Zenith in place A, what will be the length of the shadow of the gnomon at the place B which is 1000 li due North from A?

Or: from the shade length of the gnomon at a given place, find the place on Earth where the Sun is in Zenith.

All these problems assume that the Earth is flat.

As the commentators do not say anything about this, I conclude that in 7th century AD they did not suspect the Earth to be round. This shows that Greek knowledge did not penetrate to China for more than 1000 years.

Ref. https://www.researchgate.net/publication/237753793_Ancient_Chinese_Mathematics_Right_Triangles_Their_Applications

See Joseph Needham's momumental work : Science and Civilisation in China: Volume 3, Mathematics and the Sciences of the Heavens and the Earth, Cambridge UP (1959), page 212-213:

"Rather characteristically Chinese, however, was the insistence that the heavens were circular and that the earth was square, an idea which would arise naturally enough from the circles of the celestial sphere on the one hand and the four cardinal points of earthly space on the other."

And see into : Dainian Fan and Robert Cohen (editors), Chinese Studies in the History and Philosophy of Science and Technology, Kluwer Academic Publishers (1996):

• Jin Zumeng, Critique of "Zhang Heng's Theory of a Spherical Earth", page 431:

"The Ming scientist Xu Guangqi (1562-1633) for the first time introduced the Western theory of a spherical earth to Chinese academic circles."

Regarding Xu Guangqi's works:

In 1607, Xu and Matteo Ricci translated the first parts of Euclid's Elements into Chinese, introducing his countrymen to new concepts in mathematics and Western logic. Chinese scholars credit Xu as having "started China's enlightenment".

After followers of Xu and Ricci publicly predicted a solar eclipse in 1629, Xu was appointed by the Emperor as the leader of an effort to reform the Chinese calendar. The reform, which constituted the first major collaboration between scientists from Europe and from the Far East, was completed after his death.

About Zhang Heng (AD 78–139)'s astronomical theories :

In his publication of AD 120 called The Spiritual Constitution of the Universe Zhang Heng theorized that the universe was like an egg "as round as a crossbow pellet" with the stars on the shell and the Earth as the central yolk.

The theory posited by Zhang and Jing was supported by later pre-modern scientists such as Shen Kuo (1031–1095), who expanded on the reasoning of why the Sun and Moon were spherical [see Needham, Volume 3, 415–416].

The theory of the celestial sphere surrounding a flat, square earth was later criticized by the Jin-dynasty scholar-official Yu Xi (fl. 307-345). He suggested that the Earth could be round like the heavens, a spherical Earth theory fully accepted by mathematician Li Ye (1192-1279) but not by mainstream Chinese science until European influence in the 17th century [see J.Needham, Volume 3, pp. 220, 498-499].