You are mixing up several issues:
1) Desire for either a separate independent Tamil state or substantial autonomy for a Tamil area within a federal system
2) Support for the LTTE.
3) Historical questions about 'native peoples'.
Taking them in reverse order:
3) Some Tamils (Sri Lankan Tamils) have been in Sri Lanka for a long time, others came over in the 1800s (Indian Tamils)
link but this is irrelevant to everyone having equal rights in Sri Lanka or the question of a minority wanting their own separate state.
2) Both the LTTE and the Sri Lankan government have been guilty of many human rights abuses. You don't need to support either one to hold an opinion about independence/separatism.
3) You say the campaign for a separate Tamil state "isn't really a fight worth having". However while someone may be against a violent military/terrorist methods, there are valid reasons for people to want independence. Ultimately it is up to a local population to decide if they want to be independent, and whatever the outcome people (from any group in the population) should have equal rights under the law anyway.
Suggesting that only 'native people' have political rights is the kind of thing the BNP advocate. Saying that people shouldn't have independence is a bit like the English telling the Scottish what they should do - in fact it should be up to the people who live in the area effected to decide if they want to separate. Running together 'support for the Tigers' with 'Tamil independence' is suggesting that anyone who wants independence is a terrorist - transpose this argument to Northern Ireland for example to see how this isn't valid.
You might be
"totally ignorant about the Sri Lankan civil war" but you can get a long way by spending 10 minutes reading wikipedia for lots of uncontroversial facts and figures and applying your logic about 'native people' and 'independence' to situations you may understand better in the UK, to see if they are valid principles.