Both my phone and my SD card (exFAT formatted) support big file size. However,
Android File Transfer app does not allow send files bigger than 4GB. Is there a way to bypass the limitation?
According to Android.com:
Browse the files and folders on your Android device, add folders, copy files up to 4GB to or from your Mac, delete files, and more.
As a workaround, install an FTP server on your Mac, and transfer the file via Wi-Fi or mobile data. OR, via
adb pull (adb pull /storage/sdcard0/ dir here) using the Android SDK. Source.
A simple solution is to use the free app AirDroid to transfer files over your WiFi network directly to your tablet. All you have to do is install it on your phone/tablet, enable it, and then visit the device's IP address and port number from your web browser. You can then browse the device's file system and transfer files over 4 GB with no issues. The only downside is that the speed of the transfer is dictated by the speed of your WiFi network and especially your device's connection to it. Generally I have found it to take about 1.3x as long on my 2013 Nexus 7 when compared to a USB 2.0 transfer.
Hope this helps anyone who comes across this question as it is more of a layman's answer.
Connect using ethernet cable and copy it: https://support.apple.com/kb/PH18710?locale=en_US
Try using a split archive, you can make these with 7-zip (Windows), Keka (Mac), or using standard *nix utilities if you're good with the command line. By making the archives 4 GB, you can batch transfer and decompress on your phone using something like RAR, then delete the archives. Keep in mind you will be using ~double the space temporarily (depending on compression ratio, if any) so you need space first. In Keka (which judging by your tags, you will use), put "4 GB" in for the "Split:" option. Copy all resulting files after compression (the ones labeled .001, .002 and so on to your phone. Use RAR on the first file, and the file will decompress. You can then delete the archive files off of your phone.