Biasing SVM algorithm towards particular subset of data
I don't think that's a very good idea: the goal is not to make the model predict a more extreme polarity when the tweet relates to the company.
Instead you might want to consider oversampling the few instances of this specific company. For instance if you have 100 company-specific tweets and 1000 general tweets in your training set, you could duplicate the company-specific ones 10 times in order to give the specific tweets have a higher weight in the data. If possible you should tune the parameter of how many times to duplicate in order to obtain the optimal value.