late-night travel from Takayama to Tokyo

We wish to visit the Takayama festival on April 14, but our return flight leaves Tokyo on April 15 at 1300 .
We were unable to get the relevant information from google maps (there were no schedules for those dates) or HyperDia.
Is there any other site where we could get that information?
Do you have any recommendations about how to go about planning that trip?

http://www.hyperdia.com/en/cgi/en/search.html?dep_node=TAKAYAMA&arv_no...
An early start and a bit tight, but no problem getting the schedule from Hyperdia.

But the above connections get the OP to the airport only 1 & 1/2 hours prior to departure at best.

There will be a timetable revisuon between now and the travel dates.

Would be safer and easier to change flight date.

HyperDia is not yet reliable in January for train schedules in April.
Currently there is a disclaimer shown around the bottom of each search result page. :
"The train and flight schedules are available from December 1, 2017 to March 31, 2018."

On the 17th of March 2018, JR companies will revise timetables for major sections.
Now it is too early to expect precisely which train(s) you would take.
Generally HyperDia tends to be late in taking new time data.

Do you have any recommendations about how to go about planning that trip?

It takes at least 4 hours to go from Takayama Station to Shinagawa | Tokyo | Shinjuku Station .
To get to your Tokyo hotel before the midnight, you cannot be in Takayama until late at night.

According to the current timetables, you can take the last 18:46 Limited Express (WideView) Hida until Nagoya then catch the 21:24 Shinkansen Hikari, arriving to Shinagawa / Tokyo at 23:02 / 23:08.

In 2018, the two days April 14 and 15 for the spring Takayama festival will be in the weekend;
so, it may be hard to book a hotel room in Takayama for one or two nights from Friday, April 13;
buses connecting Takayama and Shinjuku might be also crowded .

our return flight leaves Tokyo on April 15 at 1300 .

From which airport will you fly out: HND or NRT?

Narita Airport, though it is located outside Tokyo, is often regarded as a Tokyo airport.
Te get to Narita Airport [NRT], you need about 1 hour more compared to Haneda Airport [HND];
if your flight departure is at Terminal 3, add half an hour.

Perhaps consider renting a car that you could drive overnight or the next morning? However, if you don't usually drive on the left side of the road, this could be risky, especially at night.

It would be cheaper to pay for an airplane ticket rebooking fee than it would be to hire a car and pay the one way fee.

Fair point, but it depends on how many people are included in the "we". I agree that OP shouldn't risk missing their flight on tight connections.

According to the Takayama City Sightseeing website

http://kankou.city.takayama.lg.jp/2000002/2000024/2001064.html

Extra limited express train Hida 98 for Nagoya will start at 20:44.

春の高山祭に合わせ、臨時列車「ひだ98号」（高山駅20時44分発）が運行されます。

Though you can't reach Tokyo on 14th, you will be able to start early from Nagoya and will have sufficient time. If you can't get on Hida 98, there will be 21:45 train for Minoota (23:59).

http://www.hyperdia.com/en/cgi/en/search.html?dep_node=TAKAYAMA&arv_no...

There are some vacant rooms on 14th April at Minoota.

Perhaps the OP want to get night bus from Takayama to Tokyo, but as far as I searched there is no such bus.

According to the Takayama City Sightseeing website
http://kankou.city.takayama.lg.jp/2000002/2000024/2001064.html
Extra limited express train Hida 98 for Nagoya will start at 20:44.

Well, currently the information on this webpage is about the past spring festival in 2017, but I suppose you can expect that a similar extra Limited Express train would be available on April 14 in 2018.
A possible problem is, you may not successfully reserve seats for all the members in your group.

Alternatively you might hope to take two Local trains to Toyama, stay overnight near the station and head to Ueno or Tokyo in the morning by Hokuriku Shinkansen.
The current train schedules:
(1) Takayama (20:50) [[Local]] (21:56) Inotani.
(2) Inotani (22:10) [[Local]] (23:01) Toyama.
(3-A) Toyama (06:37) [[Shinkansen Hakutaka 552]] (09:14) Ueno, (09:20) Tokyo.
: In a Hakutaka train, some seats are secured on a first-come, first-served basis.
(3-B) Toyama (07:19) [[Shinkansen Kagayaki 502]] (09:26) Ueno, (09:32) Tokyo.
: In a Kagayaki train, all the seats require reservation.