Example word-trivia website that calls ていとう "correct", but
https://zatugaku-gimonn.com/entry1199.html
ただ一般的には提灯は「ちょうちん」と読むことがほとんどです。 <--- 一般的に＋ほどんど, that's pretty much always...

https://ja.wiktionary.org/wiki/提灯
提 灯（ちょうちん:唐宋音　稀:ていとう）

A random online text with  ていとう in　context (fiction?)
https://kakuyomu.jp/works/16817330663886918409/episodes/16817330663890192330
すっかり暮れた後だったが、前院(まえにわ)の燈籠(とうろう)には明かりが灯(とも)されているので、手持ちの提灯(ていとう)がなくても足元は見えていた。

Results are about the same for either kanji, neither have ていとう
https://furigana.info/w/提燈
https://furigana.info/w/提灯
ちょうちん	78.9%
ちやうちん	13.4%
ぢょうちん	3.2%
かんばん	1.7%
てうちん	0.6%
あかり	0.4%
ちようちん	0.4%
ちゃうちん	0.2%
これ	0.2%
しるし	0.2%
ぢやうちん	0.2%
てふちん	0.2%
ともし	0.2%
カンテラ	0.2%
ランタン	0.2%

https://ja.wikipedia.org/wiki/提灯
提灯（ちょうちん）は、...

I see ていとう added in 2019 by Robin in the midst of some other discussion, comment "daijr, koj".  We have a lot of 提灯 entries, but ていとう is only on this one.  

Compare google results for:
"提灯(ちょうちん)"
"提灯(ていとう)"

Even for a google search for "提灯 "ていとう", if you scroll down, you get a total of 86 results before exhausting what google has. Most are irrelevant, others are reference books and vocabulary trivia websites. (first ref is an example)

"ていとう" is what one would guess.  Learners (from experience, and online anecdotes) have a habit of adopting any reading they can easily remember, until they go out in the wild and get laughed at when using it. "but the dictionary said it's an accepted reading..."   

ていとう is not in sankoku or daijs or the big ちょうちん wikipedia article.  nikk lists it under its own limited entry, referring only to a "handheld lamp".  furigana.info has zero readings as ていとう.  wiktionary lists the reading as "稀". If you search for 提灯 "ていとう", you'll get plenty of references saying "ていとう" is also 正しい, no doubt because of its presence in some venerable old dictionaries.  But it doesn't seem like using ていとう to describe decorative paper lanterns would be a good idea.

There's a pickle here, because we don't really have a "rare reading" tag, other than the ordering. It's been blessed by a kokugo (or I doubt we would include it at all).  To say it is "obsolete", implies that the reading has gone out of fashion. But when was it in fashion?  I've marked it [ok] anyway for emphasis, for lack of another option.

The "handheld lantern" gloss is merely the best I could suggest from the nikk entry:てい‐とう【提灯】
〘名〙 手にさげるようにした灯火。ちょうちん。
※歩兵操典（1928）第五六〇「夜間に在りては通常提燈を〈略〉標定点と為す」

Note the nikk source is a military manual, again suggesting a very particular sense.  Unless the soldiers were using paper lanterns in the 1930s(maybe?)

===
Since I was reading the kokugos... Even sankoku has sense [3](as does nikk, but not daisj?). Here are examples that are *not* 鼻ちょうちん:
鼻からちょうちん	813	92.2%
鼻から提灯	69	7.8%

google image search shows this a very-alive colloquialism.

Maybe   [restr=ちょうちん] on [1] is not 1000% strictly true (but look at all our collocations that do *not* include it...).  I'd welcome other options that indicate "this is here because it's in a kokugo, but please never use this".

@@ -18,0 +19 @@
+<re_inf>&ok;</re_inf>
@@ -20,0 +22 @@
+<stagr>ちょうちん</stagr>
@@ -24,0 +27,14 @@
+</sense>
+<sense>
+<pos>&n;</pos>
+<misc>&rare;</misc>
+<gloss>handheld lantern</gloss>
+<gloss>handheld lamp</gloss>
+</sense>
+<sense>
+<stagr>ちょうちん</stagr>
+<pos>&n;</pos>
+<xref type="see" seq="2714610">鼻ちょうちん</xref>
+<xref type="see" seq="2714610">鼻ちょうちん</xref>
+<misc>&col;</misc>
+<gloss>snot bubble</gloss>

Point taken.
In a fork Marcus wrote: "The yakitori sense should perhaps be a separate hiragana-only entry, like ヒモ／ひも, for example, on second thought."
I agree - I can't see the kanji form being used for that dish. I'll make that a new entry.

@@ -26,15 +25,0 @@
-<sense>
-<pos>&n;</pos>
-<xref type="see" seq="2684470">提灯記事・ちょうちんきじ</xref>
-<xref type="see" seq="1734800">提灯持ち・ちょうちんもち・2</xref>
-<misc>&id;</misc>
-<gloss>puffery</gloss>
-<gloss>praise</gloss>
-</sense>
-<sense>
-<stagr>ちょうちん</stagr>
-<pos>&n;</pos>
-<misc>&uk;</misc>
-<gloss>chochin</gloss>
-<gloss g_type="expl">yakitori skewer of chicken oviduct meat and immature egg yolks</gloss>
-</sense>

The 提灯　in ちょうちん袖 and ちょうちんバネ refers to the shape.
提灯記事 is essentially an abbreviation of 提灯持ちの記事. I think it's enough to have an x-ref to 提灯持ち on 提灯記事 (and any similar words).
I don't see a "puffery; praise" sense in 提灯 itself.

In addition to 提灯記事 and 提灯持ち, I'm seeing:
提灯売り[買い]  following (the lead of) speculators on the stock market.
ちょうちん袖 : puff sleeve
ちょうちんバネ : box spring
I feel that we should either have a separate sense, or at least get the puffery/praise meanings into sense 1.

daijr, koj

Unless 提灯 is used in other words with that meaning, I don't think we can deconstruct the word in that way.
Added reading.

@@ -15,0 +16,3 @@
+</r_ele>
+<r_ele>
+<reb>ていとう</reb>
@@ -31,0 +35 @@
+<stagr>ちょうちん</stagr>

Better? Since fig isn't a sense-wide tag

@@ -27 +27,2 @@
-<gloss g_type="fig">puffery</gloss>
+<misc>&id;</misc>
+<gloss>puffery</gloss>

GG5, etc.

No reference has this as a sense, but it occurs often in occurrences of 提灯.
Attempted trim of the food sense.

@@ -24,0 +25,7 @@
+<xref type="see" seq="2684470">提灯記事・ちょうちんきじ</xref>
+<xref type="see" seq="1734800">提灯持ち・ちょうちんもち・2</xref>
+<gloss g_type="fig">puffery</gloss>
+<gloss>praise</gloss>
+</sense>
+<sense>
+<pos>&n;</pos>
@@ -27 +34 @@
-<gloss g_type="expl">yakitori skewer of chicken oviduct meat and egg yolks from eggs taken before they have entered the oviduct</gloss>
+<gloss g_type="expl">yakitori skewer of chicken oviduct meat and immature egg yolks</gloss>

https://macaro-ni.jp/37700
google images

@@ -22,0 +23,6 @@
+<sense>
+<pos>&n;</pos>
+<misc>&uk;</misc>
+<gloss>chochin</gloss>
+<gloss g_type="expl">yakitori skewer of chicken oviduct meat and egg yolks from eggs taken before they have entered the oviduct</gloss>
+</sense>