Check if the following VMware KB article helps: VMware Knowledge Base
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In order to keep all your configurations you will need to do a stair step upgrade. If this is too complex the upgrade can be simplified significantly, but you would need to rebuild your vSphere Replication and SRM Configuration.
I am assuming your 5.5 is GA and the 6.5 target is the latest Build.
Please note that upgrade plan validation is an offering of the VMware Premier Support Service.
I cannot validate if your Databases will be supported through out this process. so will limit this to the basic upgrade steps.
1) vSphere Replication 5.5 -> 5.8.0
2) SRM 5.5 -> 5.8.0
4) vCenter Server 5.5 -> 6.0
5) vSphere Replication 5.8.0 -> 220.127.116.11
6) SRM 5.8.0 -> 18.104.22.168
8) vCenter 6.0 -> 6.0 U3g
9) vSphere Replication 6.0 -> 22.214.171.124
10) SRM 6.0 -> 126.96.36.199
12) vCenter Server 6.0 u3g -> 6.5U2
13) Upgrade ESXi 5.5 to 6.5 u3g
14) vSphere Replication 188.8.131.52 -> 184.108.40.206
15) SRM 220.127.116.11 -> 18.104.22.168
16) Use the SRM import Export tool to backup your SRM configuration
17) Upgrade VM tools on your VM's
Please note that you should follow all the best practice sections in the upgrade section in the documentation of each product. like ensuring the state of SRM preupgrade.
Please also ensure that you take backups through the process, and keep snapshots so you can roll back if an individual step fails.
The test is an important functional test and should not be skipped. This means that Both sites will need to be upgraded in parallel.
The step 17 upgrade VM tools should not be skipped as any IP customization steps will not function correctly without the newer VMware/Open tools.
These upgrade steps are my quick opinion and should be verified. Please note that you will need to consider any other products in your environment (such as database servers, Drivers, Firmware on your physical hosts/Arrays/other infrastructure components; etc).
If you require assistance with the upgrade process VMware does have a Professional Services offering. VMware Professional Services | VMware Consulting | United States