I’ve read How to easily install third-party apps, even if your carrier prohibits it – Android and Me article and decided to try installing Swype keyboard on LG Phoenix. A link to Android Injector on that article is broken. A correct link is provided on XDA page.
Swype is a more difficult case than other apps: to install Swype you need to download and install Swype Installer app and then run it to install Swype itself. You can use Android Injector to install Swype Installer, but then your phone with forbidden side-loads will not allow Swype Installer to install Swype.
I found a work-around instruction for Sideload Wonder Machine and adjusted it for using Android Injector and adb tool (adb is installed as a part of Android SDK).
Here's how to do it:
1) Sign in for Swype Beta and download Swype Installer once they respond to your request.
2) On your phone enable USB Debugging (Settings -> Applications -> Development -> USB Debugging.
3) Connect your phone to your computer's USB. Do not turn ON USB sharing.
4) Launch Android Injector, find and select Swype Installer, install it to your phone.
5) Close Android Injector, you don't need it anymore.
6) On your phone launch Swype Installer. After selecting a Swype variant you need, you will get a message telling that loading non-market applications is disabled. Don't click anything, just leave your phone connected to a PC for a moment.
7) On your PC open a command window and navigate to a folder where adb.exe is located (.... android-sdk-windows/platform-tools/ folder)
8) Run the following command:
adb pull /data/data/com.swype.android.installer/files/mypackage.apk ./mypackage.apk
9) Run the following command:
adb install mypackage.apk
10) On your phone click on "Cancel" button on the warning. Proceed normally with installing Swype.
11) Disconnect your phone, close command window on your PC. You are done.