Yes you can.
A DKIM signature looks like this:
DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=example.com;
s=selector1; h=From:Date:Subject:Message-ID:Content-Type:MIME-Version;
bh=M1X/nttSCpN...ttSCpN=;
b=SPso8U12ChySEQcnJcvm76...RAxjJFcBI=
During validation the selector and domain (selector1 and example.com above respectively) are both used to locate the TXT record with the public key. This is done by merging them with _domainkey in between:
selector1._domainkey.example.com
In your case you can use 2 different selectors (like cloudapp1 and cloudapp2), then each private key can have its own unique DNS record. The corresponding DNS entries would look something like:
cloudapp1._domainkey 3600 IN TXT "v=DKIM1; k=rsa; t=s; p=MIGfMA0GCSqGSIb3DQEBAQUAA4GNADCBiQKBgQDfl0chtL4siFYCrSPxw43fqc4zOo3N+Il220oK2Cp+NZw9Kuvg8iu2Ua3zfbUnZWvWK4aEeooliRd7SXIhKpXkgkwnAB3DGAQ6+/7UVXf9xOeupr1DqtNwKt/NngC7ZIZyNRPx1HWKleP13UXCD8macUEbbcBhthrnETKoCg8wOwIDAQAB"
cloudapp2._domainkey 3600 IN TXT "v=DKIM1; k=rsa; t=s; p=MIGfMA0GCSqGSIb3DQEBAQUAA4GNADCBiQKBgQDfl0chtL4siFYCrSPxw43fqc4zOo3N+Il220oK2Cp+NZw9Kuvg8iu2Ua3zfbUnZWvWK4aEeooliRd7SXIhKpXkgkwnAB3DGAQ6+/7UVXf9xOeupr1DqtNwKt/NngC7ZIZyNRPx1HWKleP13UXCD8macUEbbcBhthrnETKoCg8wOwIDAQAB"