Gnasher729 points out the biggest problem with splitting everything into small packets... There is a 2nd problem with this technique, which is a bit more esoteric.
Encryption techniques can be put into 2 broad categories, substitution block ciphers, and mutating substitution block ciphers.
A substitution block cipher converts a meta character (1 to N bits of information where N is the block size of the cipher) to the same meta character A -> B
every single time, which is nice be cause it makes decrypting/encrypting the message fast, and parallelizable. But, it allows your message to be analyzed and broken using statistics + apriori knowledge about the message.
A mutating substitution block cipher is a block cipher that encrypts things differently as it goes thru the message by using some, hard to reverse engineer, saved state (a CPRNG, previous cyphertext etc...) so that your meta characters map differently in the throughout the message A->B
then A->C
. This makes your message much more secure because statistical analysis is far more difficult. The downside of course is that your message can only be decrypted/encrypted as a whole message. And poorly chosen mutating saved state can make the message less secure, ENIGMA is a good example of a poorly chosen mutating cipher... Which is why you shouldn't roll your own Crypto.
You could get around this particular weakness by generating a new RSA key for every single payload. But I don't recommend that.
Edit: Realized you wanted a recomendation, which this fails to provide... This is my recomendation.
Use RSA to establish a communication channel and then send a randomly generated key using RSA.
Use that randomly generated key to encrypt the message you want to send using a symmetric encryption algorithm, like AES GCM (do not use AES CBC, it is flawed).
This is relatively easy to implement.