How to force gpg2 to use expired key?

Use the --faked-system-time option:

gpg2 --faked-system-time 20100101T000000 -e -r keyid

I would prefer an option that would force encryption to an expired key while also recording the correct time (especially when signing).

As @rob suggested in comments, libfaketime handles this well:

$ gpg  -v -v -v --batch --trust-model always -ear D5B9D38C <<< "qweqe"
gpg: using character set 'utf-8'
gpg: Note: signature key 077EA269D5B9D38C expired Sun 09 Nov 2014 12:57:25 PM +03
gpg: D5B9D38C: skipped: Unusable public key
gpg: [stdin]: encryption failed: Unusable public key

$ faketime  '2012-12-24 08:15:42'  gpg  -v -v -v --batch --trust-model always -ear D5B9D38C <<< "qweqe"
gpg: using character set 'utf-8'
gpg: using subkey 85231B396B9569ED instead of primary key 077EA269D5B9D38C
gpg: No trust check due to '--trust-model always' option
gpg: reading from '[stdin]'
gpg: writing to stdout
gpg: RSA/AES256 encrypted for: "..."
-----BEGIN PGP MESSAGE-----
...

Old answer:

1. Start UML (user mode linux)
2. Set date inside UML to acceptable range.
3. Encrypt message in UML

Reset hardware and system clock:

sudo hwclock --set --date "12/31/2017 23:59:59" && \
sudo hwclock --hctosys && \
gpg --sign-with $keyid --clearsign --armor < $file

Afterwards (I hope you have NTP enabled) fix your hardware clock with:

sudo hwclock --systohc