I think there are some misconception here with the premise of this question. There actually was a large degree of cultural influence from India to Myanmar historically. And Myanmar isn't "Sinitic", nor is there an abrupt cultural change when one crosses the border from India to Burma.

1. "Sinitic" is usually meant to denote China and Chinese. Linguistically, Sinitic is the name of the branch of Sino-Tibetan that consists of the Chinese dialects/languages. Burmese, the official language of Myanmar and the language of the ethnic majority, is a Sino-Tibetan language, but it's on a different branch from the Sinitic branch, divided by the Himalaya. You can read a quick introduction of the Sino-Tibetan language family on the wiki.

2. If by "Sinitic", you mean "influenced by China", then I think the word you're looking for is "Sinosphere", which originally was meant to denote "China from the periphery" (see: Matisoff original definition). Not really a scientific term nowadays, but in the modern day, it's usually understood as "countries that have had a history of using Han characters" or "chopstick countries" colloquially, namely the modern day CJKV group (China, Japan, Korea, and Vietnam; the category can be technically larger if you count states that no longer exists as independent political entities such as Jurchen Jin, Khitan Liao, Nanzhao, Dali, Ryukyu, etc.). A common point of this group is that they all historically used Hanzi and officially followed a syncretic world view of their respective folk religions, East Asia Buddhism (mainly Chinese-style Mahayana), Taoism, and Confucianism. Meanwhile, Myanmar has never had a history of using Hanzi as the official writing system, did not follow Confucianism, did not follow Taoism, has had Hindu-Buddhism, then non-Chinese-style Mahayana Buddhism, then Theravada Buddhism as their official religions. In any case, the ethnic groups of Myanmar does have some Chinese influence, either historically or from recent history of the 15th-20th century, and shares some common cultural points with the Han Chinese, but they're usually not considered a "Sinosphere" country.

3. For some context of Myanmar's peopling history, the land where Myanmar is now was initially inhabited by the Pyu, Mon, and Wa people. Before the Bamar (the modern ethnic majority of Myanmar) came, the Pyu, Mon, and Wa people all built their own kingdoms and city states. Around the 9th century, Nanzhao kingdom (Yunnan) collapsed and was replaced by Dali kingdom, sending refugees running south to SEA. The Bamar was one of these groups and arrived at Myanmar, defeating and absorbing the kingdoms of Pyu, Mon, and Wa, driving the Wa north to the border with Yunnan (also their modern location). As they absorbed these kingdoms, they also absorbed the local Indianized culture and Hindu-Buddhist religion of their predecessors. In the 10th century, the Tai-Shan people, also originally from Yunnan, came down south and adopted the local religion and culture. Lastly, from the 15th to the 20th century, Han Chinese started to make their way to Myanmar for either refugee from wars or for economic reasons. Most of Myanmar's Chinese influence came from these 15th-20th century waves of Han Chinese, a much shallower time-depth and scale than the Chinese influence in Japan, Korea, and Vietnam.

4. In fact, among the mainland Southeast Asia countries, Myanmar as the Pagan empire and the Toungoo empire was actually the second filter through which the rest of mainland Southeast Asia got most of their Hindu-Buddhism and the later Theravada Buddhism influence from, second chronologically to the first filter of the Khmer empire. As the 2 major empires of mainland Southeast Asia at the time, the Khmer empire and the Pagan empire did most of the spreading of Hindu-Buddhism during the 9-10th century period through their influence on the nearby people living along maritime trade routes and through their conquest. In the 15th century, the Toungoo empire at their height went conquering all over mainland Southeast Asia, bringing along with their influence Theravada Buddhism to the rest of the mainland SEA (except for Vietnam), replacing the Hindu-Buddhism and Mahayana Buddhism of the previous era.

(tbc)