You will need to do what FL DMV requires to get a valid license there. If you do not comply with FL DMV than RI will not grant you a valid license.
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I do not practice in FL, so I am not sure what the effect of not using the ignition interlock system will have on you in FL (i.e., whether it means you will not get a valid FL license reinstated or whether it would result in a probation violation), and while RI does have a relatively new ignition interlock system statute, I do not see how RI itself can impose this requirement on you, especially if you are not applying for a RI driver's license. What would happen if you do not have a valid FL license and are operating a vehicle in RI is that you would be charged with a misdemeanor for driving with a suspended/invalid license. Again, I would certainly check with a FL lawyer to see what impact your refusal to use the system would have on your conviction or your FL license. If you are thinking of applying for a RI license, it is possible that they may not give you one unless/until you have completed the conditions of the FL courts or DMV.