Hello and welcome to the communities.
The license keys between v3 and v4 are not compatible, so you won't be able to assign a v4 key to the v3 ESXi. The licensed version would give you access to the backup APIs and solve your problem though.
Do you just have a single ESXi host?
Yes, we have only one ESXi host.
- Do you know of any workaround or any other means to take backups "on the fly" from ESX3.5i?
I followed the link you provided, it seems to be a collection of scripts for host maintenance.
But I can't find anything related to my questions there, probably due to my lack of knowledge.
- Can you give any other hint about a utility or method that can take backups on running vms from a free ESX3.5i?
I've tried Veeam Backup & Replication 6.5 and it works fine when copying files from shut down vms, but when I try to "VeeamZIP" a running VM I encounter the "RestrictedVersionFault" problem.
I should have been more specific with that link. Look on that page for the "ghettoVCB" scripts. Those are backup scripts.