Go to the Conservative Party. Literally quit the Republican Party and re-register as a Conservative. You are still free to vote Republican and indeed, in most elections, the candidate will have the endorsement of both parties. In my experience, however, a strong conservative party will move the Republican Party to the right if only so that the Republican candidate retains the Conservative endorsement.
Also, do not give money to the Republican Party or its numerous PACs at the national, state, or local level. Either contribute financial support directly to the candidate or donate to the Conservative Party. (While your local Republican Party has no control over national immigration policy, you can bet that the National Republican leadership will get an earfull if people stop funding the local party because of the immigration issue.)
In addition, in many states, a party's ballot line is determined by the number of votes the party received in the last govenor's election. Every party obviously wants ot be on the top line, and certainly no worse than second. If enough people vote for the Conservative candidate, even if he or she is also the Republican candidate, then the Conservative Party would appear above the Republican line of the ballot and that would send a very strong message.
Finally! A common sense idea that's not going to shoot ourselves in the foot. Thank you so much Labyrinthos for your clear thinking! I will look at their website immediately. Any links/information/tactical insight that you have on the conservative party would be appreciated.