The ELISAs are certainly convenient (if you have a robot or student to do them). Whether they're sufficient really depends on what your protein is supposed to be doing.
For example, if it's a vaccine, or if it's supposed to work in conjunction with a vaccine, you may want to look for antibodies against your target. Or if it's supposed to do something to immune cell cell multiplication or migration, then a complete blood count might be in order. So maybe try to tell us a little bit more about which specific component of the immune system you're trying to stimulate and how it's supposed to function.
I just want to verify if a fused protein construct could stimulate an unwanted immune response. It will be just a little part of my project so I want a standard but sufficient experiment ( it will not be a vaccine but more like a drug )
If you consider your fused protein construct an immunogen, then you could measure T cell based responses as you mentioned by cytokine release (https://www.iqbiosciences.com/bioservices/safety_assessment/cytokine-release-assay/) or by measuring antibodies that are generated against it after repeat injection into an animal for example. These are things that we routinely monitor for biologics as part of our research level safety assessment for unwanted Immune Responses at iQ Biosciences.