思うて - is that like 思って, but old timey or dialect?

I'm reading a fantasy novel and this came up:

ヤムル帝の妃と名のった方は、ふしぎな話をなさいました。そして、くれぐれも夢と思うて忘れるでないぞ、しかと帝にお伝えせよ、と、おっしゃったのです。

I'm confused by the second sentence. I think it's talking about what the Kisaki said.
しかと帝にお伝えせよ - convey this to the Mikado
夢と思うて忘れるでないぞ - something about not forgetting that this is a dream? Is the Kisaki saying that, or is the speaker saying don't forget that I'm telling you about a dream?
思うて - is that like 思って, but old timey or dialect? Not really sure what this is doing in the sentence at all.

Overall my best guesses are:
Don't forget that this is all a dream, but she told me to convey this to the Mikado.
She said to not forget that this is a dream and to convey it to the Mikado.

Thanks,
Cala

くれぐれも夢と思うて忘れるでないぞ and しかと帝にお伝えせよ are both what the empress said in the dream. While ～でないぞ is a strong negative command from superior to subordinate, the speaker uses polite forms(ました, のです) to the listener, so this can't be the speaker's words.

思うて is originally a classical form of 思って, and still used as dialect mostly in Kansai region, but it's used to show the speaker's (= empress's) dignity there as same as other classical expressions. This -te form 思うて/思って is a continuous usage or expresses the reason of the following clause, thus, it's not the object of 忘れる. The object is what she said in the dream (or another interpretation "to convey it to the emperor" is also possible). ～でないぞ negates the whole preceding part, not just 忘れる part, thus, the construction is [夢と思うて忘れる (= you think this is just a dream, and/so forget it)]でないぞ. The empress actually said "Don't forget what I said(or to convey it to the emperor) because of your thought that this is just a dream." Incidentally, the object of お伝えせよ is also the dream itself. She said "Convey surely what I said in this dream to the emperor".

Thank you for your excellent explanations as always. Looking at the politeness level really does help like that.

I've come across a different problem though.

"なるほど。とてつもなく、ふしぎな夢だのぅ。……そなた、それをすべて、おぼえていたのか？
夢を、そのように、はっきりとおぼえておるものかの"

In the last sentence here, I'm struggling to grasp the tone. Is there any way of telling whether the speaker is saying: and are you one to remember dreams that clearly? (If that's correct) In a surprised or dismissive way?

thanks as always,
Cala

もの is not 者(person) but 物(thing) there. ～かの is an old/noble people's version of ～かな, and そんなに～かの/かな shows the speaker's doubt. Have you ever seen expressions such like そんなにうまくいくものかな（＝そんなにうまくいくとは思えない） or 寿司ってそんなにおいしいかな（＝そんなにおいしいとは思えない）? The speaker is a bit skeptical about the dream/what he heard.

Yeah, I was suspecting the skepticism because I've seen expressions like that. But since the mono didn't come with kanji I couldn't work out if it was the expression or the person mono. As always many thanks.