It is certainly feasible to do so given modern processing power, but it is bad idea for several reasons.
Firstly, you can authenticate the server with a certificate, but what about the client? You could either use client certificates - problematic to manage, or you could generate a key pair at the client every time (slow) and authenticate the client with a password after establishing a secure session. AES on the other hand, just needs a short key and comes with support at the instruction set level from modern x86 and AMD processors.
Second, and judging from the fact that you're looking at RSA because you need a 'very security-intensive solution', you should know that RSA-2048 is actually less secure than AES-128. 2048 bits of RSA key is equivalent to about 112 bits of symmetric key strength. See here.
Lastly, the key exchange problem in symmetric schemes like AES is typically solved using an asymmetric scheme like RSA or DHKE. So what you should be doing is using a public crypto scheme to exchange symmetric keys, and then using a strong symmetric cipher for the rest of the data - this is exactly what TLS does.
HTTPS is the most well known application layer protocol - it stacks HTTP on top of TLS. TLS is widely supported, OpenSSL is probably the most popular library in use today, and is supported across all major platforms, use it.
Edit 1: Banks don't just apply SSL security because they realize that SSL doesn't solve everything - there are problems that it's not meant to solve. For example, it is useless against phishing or session stealing. The objective of the bank's authentication mechanism is not to check if the person it's talking to knows the password - it's to check if the person is You. This means also making sure the connection is coming from a recognized IP address, sometimes making you answer Security Questions, and asking you to make sure you recognize the bank website by using what are called SiteKeys.
Edit 2: You cannot make sure that the information isn't intercepted by a third party. The best you can hope for is that they cannot distinguish your data from a perfectly random stream of bytes, and this is the goal of every network security protocol. Some schemes also try and hide the actual source and destination addresses, such as IPsec, but this is uncommon.