Short Answer:

A timely question, as in 2013 UNESCO put native Japanese cuisine, washoku, on a list of cultural heritages, leading to scholarly inquiry into the question of just how much of the cuisine is foreign influenced. While there's a great appeal to a hard distinction between washoku and yōshoku (the Japanese term for western cuisine), in practice the two are much more mixed than is often acknowledged. Perry's "opening of Japan" didn't change much as an event, but the subsequent changes during the Meiji Restoration did.

Discussion:

Japan has always been a nation of foreign influences, refashioned into distinctive local products, and this is true of cuisine. China and Korea are obvious long standing influences, but in terms of Western cuisines, it's the Portuguese who have the most dramatic legacy.

Both tempura and sukiyaki are derived from frying techniques learned from the Portuguese at the trading port of Nagasaki; long before Perry.

It's not so much Perry and his limited and unwelcome "opening of Japan" that transforms Japanese culture in the second half of the 19th century, it's the Japanese response to it, the Meiji Restoration. The Japanese send missions around the world to learn foreign technologies, bring in foreign experts to teach science, technology, medicine, and many other aspects of modernity. They enthusiastically adopt both as policy and culture

It's in this process that you get new culinary and distilling technologies in Japan-- today, for example, Japanese whisky fetches some of the highest prices in the world; the first Japanese whisky was distilled in 1870. While Japan had somewhat similar drink shōchū, whisky is distinct, and was learned from the example of Scotch whisky. Shōchū itself seems to have arrived centuries earlier, from Asia.

The 1870s and Meiji bring in Western influences on elite dining, but on the popular level you find yōshokuya - small restaurants serving western cuisine.

serving

an inexpensive, Western-style cafeteria that provided the Japanese public with domesticated versions of selected Western-style dishes, such as roast beef, roast chicken, beef steak, veal cutlet, croquette, curry on rice, omelette and stew (Maenobō 2000: 97). The reliance on Anglo-Saxon cooking rather than French cuisine was determined by two factors. First of all, it was representative of British and American supremacy in the life of the Western communities in the treaty ports. Because the cafeterias were being set up and run by former employees of Western households or restaurants that were patronized by the overwhelmingly British and American community in Japan, they featured food that had been prevalent there. The second reason was the fact that English and American dishes were much less complex and therefore easier to prepare

As anyone whose traveled or studied in Japan has likely experienced, the Japanese typically process foreign innovations into their own distinctively Japanese forms. Karē-raisu the ubiquitous "curry rice" is a distinctly Japanese take on Anglo-Indian cuisine, distinctive from anything either Indians or British eat.

Similarly the Japanese adopted Worcestershire sauce as a ubiquitous condiment in a way that it isn't in the West, in Japan it became a sort of "European soy sauce", used far more than in Western cuisine.

Interestingly there are often cues in Japanese orthography to foreign food origins, -- Japanese typically use katakana to spell out foreign words; and that habit allows us one measure of "how Japanese Japanese think it is". So, for example, you typically see ビーフステーキ -- "beefsteak" -- on a menu written in katakana, same with karē-raisu, ( カレーライス ) .

Tempura is an interesting case, sources have it that the word "tempura" derives from quatuor anni tempora -- the times when Catholics don't eat meat or may fast, "Ember Days" (though I've yet to see a primary source for this, there's a risk of a just so story here). Whatever the etymology, "tempura" walks the line of acculturation: you will see it written in katakana as テンプラ; but it also gets transformed into its own Japanese character and hiragana, or even all Japanese characters, as 天ぷら or 天麩羅. Notice that the first character "天" -- read "ten" is the character for "heaven" and also the first character in the Emperor (Tenno)'s Japanese orthography (天皇); an orthographic evidence that Japanese treat this as "their own". Not fully washoku, perhaps, but getting there.

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Sources:

[Katarzyna Cwiertka, Chair of Modern Japanese Studies at University of Leiden, has made European influences on Japanese food and dining styles a speciality and written extensively on the topic] https://www.cwiertka.com/

CWIERTKA, KATARZYNA J. “EATING THE WORLD: RESTAURANT CULTURE IN EARLY TWENTIETH CENTURY JAPAN.” European Journal of East Asian Studies, vol. 2, no. 1, 2003, pp. 89–116.

Cwiertka, Katarzyna J. “Serving the Nation: The Myth of Washoku.” Consuming Life in Post-Bubble Japan: A Transdisciplinary Perspective, edited by Katarzyna J. Cwiertka and Ewa Machotka, Amsterdam University Press, Amsterdam, 2018, pp. 89–106.

Cwiertka, Katarzyna J. (2006) Modern Japanese Cuisine: Food, Power and National Identity, London: Reaktion

Ishige, Naomichi (2001) The History and Culture of Japanese Food, London: Kegan Paul.

Rath, Eric C. and Assmann, Stephanie (eds) (2015) Japanese Foodways, Past and Present, Urbana, IL: University of Illinois Press.

Cwiertka, Katarzyna J. (2018) "Washoku, heritage and national identity" in Routledge Handbook of Modern Japanese History, Routledge.

Mary Redfern (2014) Getting to Grips with Knives, Forks and Spoons: Guides to Western-Style Dining for Japanese Audiences, c.1800–1875, Food and Foodways, 22:3, 143-174,