In Lightroom on iPad, which method did you use for exporting?
Share > Share To
Share > Open In
Share > Export to Camera Roll
Share > Export to Files
Share > Export As
When I tried it using Export to Files, the caption and keyword metadata is present, and:
Not shown in iOS Files > Info
Shown in the EXIF Metadata app for iOS (very useful app by the way)
Shown in Affinity Photo for iOS in its Metadata Studio > File panel
Shown if transferred to a computer and viewed in Properties or Adobe Bridge
So one of the things you may be seeing is a limitation of metadata display in the Info panel in the iOS Files app.
Note: The Tags category in iOS Files is for Apple Tags only. Keywords added in photo applications do not show up in Tags, because Apple Tags are not compatible with IPTC keyword tags or any other standard metadata. Apple Tags are for general organization of any file type across Apple desktop and mobile devices.
Also, just to make sure: When I exported from Lightroom on iPad, the Export to Files settings had Include Metadata enabled as shown below. If Include Metadata is disabled, the metadata will not be in the exported file.
Does any of this help narrow down the problem? Because it looks like it should be working if set up properly and viewed properly.