Dumping the contents of memory will not help you here if the software is even vaguely smart about the proper use of public key cryptography. If you need to forcibly dump your memory though, there's a handy answer on this question: How do I create a memory dump of my computer freeze or crash?
Public key cryptography makes use of asymmetric encryption, where one half of the key is used to encrypt a message and you must use the other half of the key to decrypt it. You cannot use the same half of the key to decrypt a message (or file) that was encrypted using that half of the key.
You can use a public key to decrypt a message made using the private key, or use the private key to decrypt a message created using the public key, but not private-private or public-public.
From the Cryptolocker Wikipedia page:
When first run, the payload installs itself in the user profile folder, and adds a key to the registry that causes it to run on startup. It then attempts to contact one of several designated command and control servers; once connected, the server generates a 2048-bit RSA key pair, and sends the public key back to the infected computer...
The payload then encrypts files across local hard drives and mapped network drives with the public key.
Because you only have one half of the key all that you can do is encrypt messages (files). You need the other part of the key to do the needful and recover your files.
In this case dumping the contents of memory will not be useful to you, because all it contains is the way to continue making things worse.
Your computer never holds both parts of the key, except after you have been given it.
To further elaborate...
One problem with public key cryptography is that due to the larger key sizes using it is computationally expensive compared to symmetric key (reversible) encryption. For this reason many systems use public key cryptography to securely exchange a symmetric key which is then used for further communication with lower overheads.
In this case though the use of the simpler symmetric key is unnecessary and would work against the malware author. If they used a symmetric key then you could, as you surmise, simply force all the memory to disk and start rubbing blocks of memory at your encrypted files until they open up. This is still going to take a long time though and I suspect would be infeasible given the amount of memory to check for keys. By eschewing a symmetric key stage they increase their impact at the cost of higher computational requirements.
Once the malware has started then you've already lost at least some files, and by being selective about the file types and file sizes they target they can do maximum damage with resources available. Even lower power modern CPUs could probably get a good amount encrypted before you'd notice, even with the more expensive asymmetric encryption.
By using public key cryptography they are ensuring that you need them to give you the unlocking key. Without them giving it to you there is nothing you can do.
