You've got an alpha channel. You've got to use a codec that supports alpha channels or your animation won't have transparency.
And depending on the horizontal/vertical dimentions of the comp and the frame rate, the file size could be just abbout right.
***Have you tried Dynamic Link yet?
Have both After Effects and Premiere Pro open. In the AE Project tab, select the Comp that contains your typography and choose File > Adobe Dynamic Link > New Premiere Pro Sequence.... You'll have a new linked Sequence in Premiere Pro that you can insert into your edit just like you would a rendered movie.
Be aware that there are some trade-off here. You're avoiding the large file size of a high-quality movie file with transparency, but you're not rendering yet. Also, the After Effects project file is now source media in the Premiere Pro project file.
***Are you using the "Best Settings" Render Settings and "Lossless with Alpha" Output Module in the AE Render Queue on Windows or Mac?
If on Windows, try modifying the Output Module so that you're rendering to a QuickTime movie using the Animation CODEC with the Video Output Channels set to "RGB + Alpha". You're then matching the Mac template and you'll see a good decrease in rendered file size over the Windows default.
***RGB+Alpha via Media Encoder
Presets "with Alpha" are few and far between in Media Encoder. That's partly because most compression types and/or encoding methods do not support an alpha channel. Although, even for CODECs that do, like Animation, the "Codec Settings" don't allow you to get to the color depth of "Millions of Colors+" (the plus being the Alpha Channel) as you might expect to be able to do; however, choosing "32 bit" as the Bit Depth does include the Alpha (8 bits for Red, 8 bits for Green, 8 bit for Blue, 8 bits for Alpha).
So, yes, you can export RGB+Alpha in Media Encoder. Tip: Start with a copy of the the QuickTime > Match Source (Rewrap) preset.
***Large File Sizes
When you're setting up your Comp, always check the message next to "Resolution". AE tells you how large your file will be per frame. For example, 1920x1080 is about 8MB per frame at 8 bits-per-channel. There's a lot you can do to get a smaller size when you export, but those choices depend on your workflow.