Balak saw something, and therefore Moav was afraid

Balak begins:

ב וַיַּרְא בָּלָק, בֶּן-צִפּוֹר, אֵת כָּל-אֲשֶׁר-עָשָׂה יִשְׂרָאֵל, לָאֱמֹרִי.	2 And Balak the son of Zippor saw all that Israel had done to the Amorites.
ג וַיָּגָר מוֹאָב מִפְּנֵי הָעָם, מְאֹד--כִּי רַב-הוּא; וַיָּקָץ מוֹאָב, מִפְּנֵי בְּנֵי יִשְׂרָאֵל.	3 And Moab was sore afraid of the people, because they were many; and Moab was overcome with dread because of the children of Israel.

and Rashi comments:

ב) וירא בלק בן צפור את כל אשר עשה ישראל לאמורי -
אמר: אלו שני מלכים שהיינו בטוחים עליהם לא עמדו בפניהם, אנו על אחת כמה וכמה! לפיכך ויגר מואב:

Balak… saw all that Israel had done to the Amorites He said, “These two kings whom we relied on could not resist them; we certainly cannot.” Consequently, “Moab became terrified.” - [Mid. Tanchuma Balak 2, Num. Rabbah 20:2]

As the various supercommentaries note, it is irregular that Rashi would write that 'Consequently, “Moab became terrified”' given that the next verse gives a different reason for Moav becoming terrified, namely "because they were many." All sorts of fanciful answers and harmonizations are possible, such as that they saw in the fight with Sichon and Og that Israel did not lose any people, such that they were still many, and so on.

But I would note that if Rashi's source is indeed Midrash Tanchuma and /or Bemidbar Rabbah, the bit about "consequently" might follow in a rather straightforward manner. In Midrash Tanchuma, the text reads:

משל למה הדבר דומה?
למלך שהושיב שומרים לשמרו מן הגייס, והוא בטוח עליהם שהיו גיבורים. עבר הגייס והרגם, והיה מרתת על עצמו.
ואף כך בלק ראה מה נעשה בסיחון ובעוג שהיה מעלה להם שכר לשמרו והיה מתיירא מעצמו. ועוד שראה נסים שעשה להם הקדוש ברוך הוא בנחלי ארנון
ויגר מואב

where the last two words I quoted, vayagar moav, actually begin the next section, analyzing those two words. Meanwhile, in Bemidbar Rabbah, which is later, the same text appears, but as one commentary notes, there is an extra word כתיב preceding the words vayagar moav, that should be taken out. This extra word would connect the analysis of what Balak saw with those two words, making it seem like this is the conclusion of the previous. As such, rather than asking what is bothering or motivating Rashi, we might say that Rashi is channeling his source, and the question is then shifted to what motivated the source. And then we can simply answer that it was a girsological error. Alternatively, it was a conscious choice and development of the ideas therein, to stress that what Balak saw inspired fear.

For this is certainly the peshat in this pasuk, regardless. We would not be informed for no reason that Balak saw what the Israelites did to the Amorites. That sentence has meaning in context, and only with that context do we have peshat. Surely what Balak saw caused some consternation on his part, and the fact is that it is juxtaposed with the fear of Moav in general. So various factors could contribute to the general worry. Or as Rashi appears to present it, in my opinion, Moav would not have been worried had they had the backup of Sichon and Og. Once these two fell, they were no longer confident. And then, the size of the Israelite population terrified them.

What specifically about this pasuk, or the context, provides the specifics of Rashi's comments, that they were relying on Sichon and Og? Well, this is midrash, so we might simply ask this of the midrash. But regardless, there is the issue of context. Balak saw, and we get no explicit reaction from him within the pasuk, so the juxtaposition with vayagar moav for an apparent different reason tells us that it was not just the size, but the background of Israel's success against the Emorites.

I would also note a slight shift in tone between the midrash and Rashi, where Rashi adds anu al achat kama vekama, and takes out the explicit reference to Moav having paid money to Sichon and Og to protect them. This them becomes less the idea of the loss of the bodyguards, and more the idea of that those other nations were stronger and should have been able to repel the Israelites, whereas the strength of Moav was much less. This shift -- how Rashi transforms the midrash -- might also be said to go along with Rashi's connecting it more explicitly to vayagar Moav in the next verse, for now they were unsure because of their relative national strengths.

What about the specifics of the details of what Balak saw -- according to Rashi, Sichon and Og, and according to the midrash, Sichon, Og, and as an additional possibility -- the events at Nachal Arnon? This would again be context, that all these accounts immediately precede the pasuk that about what Balak saw, at the end of the preceding perek, perek 21. Also, Sichon was clearly the Emorite, and Og well.

What about their superior strength, and relying upon them? Ibn Ezra reads this as peshat, and tries to get it from the definite article of laEmori. Since there is a kamatz there, it is a patach promoted via compensatory lengthening because the aleph would not take a dagesh. But without the definite article, there would have been a sheva there instead. And thus it refers to the known entities, the known Emorites, and those would be Sichon and Og, known from revious chapter.

I placed below Ibn Ezra a quote from the Karaite scholar Aharon ben Yosef and his supercommentary, that similarly note the heh hayidiah on laEmori but say that this is the known Emorites on this side of the Yarden, rather than those in Eretz Canaan, on the other side of the Jordan River. This would then refer to Sichon and Og (and their nations).

I think that Ibn Ezra has in mind to read Rashi's midrash into the peshat. Besides the business with laEmori, he then goes on to show that Sichon and Og (and their nations) were strongest of all the nations, such that they might be relied upon. To this end, he cites a pasuk in Amos, which in context refers to the Emorites conquered before entering the land of Israel. Thus, in Amos 2:

ט וְאָנֹכִי הִשְׁמַדְתִּי אֶת-הָאֱמֹרִי, מִפְּנֵיהֶם, אֲשֶׁר כְּגֹבַהּ אֲרָזִים גָּבְהוֹ, וְחָסֹן הוּא כָּאַלּוֹנִים; וָאַשְׁמִיד פִּרְיוֹ מִמַּעַל, וְשָׁרָשָׁיו מִתָּחַת.

9 Yet destroyed I the Amorite before them, whose height was like the height of the cedars, and he was strong as the oaks; yet I destroyed his fruit from above, and his roots from beneath.

We have this idea of exceptional strength, such that they are compared to cedars. It is easy to see how the destruction of these strongest could impact the morale of the other nations. Indeed, in Mizrachi (right), in explaining Rashi, he cites the very same pasuk from Amos.