Hey!!
Sorry I only just saw this post! Although you’ve probably written the essay I still thought you’d maybe still be looking for possible themes to look at.
Please excuse how rusty my answer may be - I haven’t really looked at my notes in a while and have holiday brain haha
I think for me I’d firstly focus on what the specific contexts were- so Elizabethan England vs 1990s America/ New York. This presents one prominent point of discussion imo. Theocentric society vs secular society.
You could really flesh out the significance of religion in RIII and what impact it has on the characters / impact of being religious or non religious.
Then you could discuss the evolving nature of religion and lack thereof in the secular New York. Impact of this? Audience effect etc.
Then also with context you could talk about evolving meaning of the text / challenges with contemporary audiences resonating with and understanding the play e.g. Replacing G with a C in LFR so contemporary audiences could easily see the proprophecies related to Clarence. You could also discuss what this means in terms of integrity of the play and text.
Also could talk about accessibility of Shakespeare - which classes / cultures have previously accessed it and why? What barriers has this created and how has Pacino and LFR tried to combat this?
Hope this helps! These are ideas I would add but it’s really subjective

Also I recommend getting a State Library card so you can access the eresources as they have fantastic critical articles - even comparative ones on the two texts!!