No. LSPs must be greater or equal to core release.
If you're on 7.1+, preferably current, and you use SDM right, you can upgrade the LSPs remotely and flip'em. Depending how many you have and how far apart they are, it might be worth looking into or it might just be worth doing them by hand.

If you're on S8300Ds, may as well spring for Es as R7 is the end of the line for the Ds anyway and if you put the spectre/meltdown patches, the s8300D's BSM capacity drops from 500 to 250 if you happen to have a lot of SIP stations. Not that spectre or meltdown would ever matter on a dedicated server like a S8300, but if your org has a checkbox for "are all my servers patched for this", you'll have to swallow a capacity hit if you're SIP phones.

SMGR 7 is backwards compatible with 6.3 however there are some significant caveats and some workarounds which may be required. For example, there are a couple of PSN's addressing certificate issues. We found once the initTM fails the ASM will accept updates to the Trusted Certificates so the System Manager certificate can be "pushed" to the SM from the SMGR and another initTM will then pass. See PSN 3952 and 5354. Additionally Session Manager 7.x below 7.0.1.2 do not seem to correctly integrate to SMGR 7.1.3 and you would patch them to 7.0.1.2 or upgrade them to 7.1.x.

Keep in mind at this point you should be upgrading to 7.1.3 or higher since many of the security vulnerabilities such as Spectre/Meltdown require upgrading to mitigate.