Did the House of Representatives during Meiji Era exercise their powers?

In short: yes.

Emperor Mutsuhito (posthumously known as Emperor Meiji) ascended to the throne at a very young age (14 years old) when his father Emperor Osahito (Kōmei) died unexpectedly of smallpox (aged 35 years). Note that due to differences between the Japanese calendar system, these ages are sometimes misreported.

As was the custom at the time, a young emperor ruled only in name. He likely performed most of his duties based on the recommendations of his advisers and cabinet. After all, these were also the regional leaders who played a role in restoring the Emperor to power and disbanding the shogunate. Notice that most of the major changes to Japan and the Japanese government occurred due to policies enacted during the early years of the “Meiji era” while the Emperor was still a teenager. It is widely accepted that Emperor Meiji ruled with a cabinet that played an important role in his new Western inspired government.

However it may be difficult to identify what specifically the cabinet did. The Japanese hold the Emperor in high regard and would attributed to many of the accomplishments during this period. This is especially since many of the changes implemented in what would have been a major upheaval at the time are now retrospectively seen as contributing to Japan becoming the industrialised regional power it is today. The few documents surviving contrary to this “great man of history” narrative are unlikely to have been translated by Japanese people into other languages.

Therefore it is unsurprising that specific examples of the cabinet executing their power are hard to verify. They likely played a major role in the “Meiji Restoration”. However, they may not have may not have to exercise their position directly as the young inexperienced Emperor would not have disputed the recommendations of the advisers that put him into power or made him popular with the Japanese people. Once taking their advice was showing results, he likely embraced their approach.

Context: The Meiji period refers to Emperor Meiji's reign, from 1868 to 1912. The Japanese parliament is now known as the National Diet - a bicameral legislative body. During the Meiji era, the parliament was known as Imperial Diet (帝国議会 Teikoku-gikai). The House of Representative is the lower body.

The early years after the creation of the Imperial Diet in 1890 is occasionally known as the "rise of political parties" because of significant political struggles due to many factions controlling the Diet. There are many reasons for this political infighting at the Diet during this initial period, but a principal one was the fact that the government of the day was controlled by the powerful Meiji oligarchy, a remnant of the Tokugawa shogunate.

Question:

Did the House of Representatives during Meiji Era exercise their powers? ... the House of Representatives were granted some form of power, such as the control of the budget which allowed them to use it as a form of leverage against the House and Peers and the Emperor, and also the ability to turn down bills proposed by House of Peers. However, I could not find any instances where they actually exercised that power.

Yes, they did. Endlessly. Between the Representatives and House of Peers, though not always in concert, they obstructed the government (nominally representing the Emperor) on many instances.

As just one instance, I will refer to a proposal to increase land tax during Matsukata's second term (1896 – 1898) as Prime Minister (emphasis mine):

The Second Matsukata Cabinet and The Land Increase Question

We saw (earlier) how at the launching of the post-war reconstruction plan Matsukata, the Minister of Finance, affirmed that land tax would not be increased. When we reached the eleventh Diet, however, two years later, the same Matsukata resolved to increase land tax, and as a result lost the confidence of the House of Representative and was forced to tender the the resignation of his government.

Note: Matsukata also held the finance portfolio during his second term.

Source: Banno, J. "The Establishment of the Japanese Constitutional System" (Nissan Institute/Routledge Japanese Studies), 1995, p. 123.