I'm not really sure we can apply the concept of "state religion" to Imperial Tibet. While Buddhism thrived during this period, many court ministers and nobles remained faithful to their Bön beliefs throughout. Even at the height of Buddhist power, powerful court ministers were still Bon believers. This would prove instrumental.
As you noted Buddhism reached new heights in the early 9th century, when they found in Emperor Ralpacan a ruler eager to pamper Buddhists. In addition to great wealth, Ralpacan even gave monks high political offices, causing widespread resentment. This eventually led to a vicious reprisal starting from ~840, as the traditional Bon powers pushed all the way back and suppressed Buddhism for over a century. Heavy devastation was inflicted as temples and scriptures were destroyed and monks killed.
After the fall of the Tibetan empire, Buddhism was said to have been preserved in Kham. It took over a hundred years before it re-established a strong presence in Tibet, when monks once again brought Buddhism to the Tibetan heartland of Ü-Tsang.
(Below was written in response to the original question)
Would it be reasonable to assume a causal link between the fall of their empire and the peak of Buddhism?
You could say that, but kind of in a somewhat convoluted way. The immediate trigger in Tibetan Empire's fall was the assassination of its last emperor, Langdarma. His death plunged the empire into a succession crisis and an accompanying civil war, leading to the Era of Fragmentation.
There's a link to Buddhism because the regicide was supposedly carried out by a monk, Lhalung Pelgyi Dorje. He was dissatisfied with the Tibetan government's oppression of Buddhism. The anti-Buddhist policies were in turned enacted by court ministers of the Bön faith, either under Langdarma's direction (traditional account) or merely using him as a figurehead (revisionist account).
These ministers were traditionally held to have assassinated the previous emperor Ralpacan. Ralpacan was a great patron of Buddhism over the period you mentioned in the question. As the story goes, this irked his Bön following ministers which ultimately led to his death and replacement by Langdarma. Though some have argued that he died of natural causes/illnesses/accidents, in which case his ministers merely exploited the opportunity to turn the table on Buddhists.
In this way the peak of Buddhism led to the disintegration of the Tibetan Empire, though how much of a link that really is would be up to individual opinion I suppose.