Thanks for the additional information. It is helpful.
Chief as you know is the alpha dog in the house. The problem is that being a small dog, if he comes up against a larger dominant dog who is living in the house he may end up getting seriously hurt as a result if the larger dog doesn't back down.
What you need to do with him is firmly establish yourself as the boss. As the boss, you will make the decision as to what dogs are allowed to do what and you will not allow one dog to attack another dog. You can correct this with formal obedience training since that helps you establish yourself as his boss. Submissive dogs do not reprimand the boss. Each time a dog obeys a command even if for a treat, they become a little more submissive to the person giving the commands. Now it doesn’t have to be a formal training class but does need to be formal training.
The following site is helpful in helping owners train their dog. Be sure and click on the link to the page on obedience at the bottom. and links on subsequent pages leading to detailed instructions.
Training works best if you train at least 30 minutes a day (two 15 minute sessions). I would start making your dog work via the Nothing in life is free program (NILF). It is outlined below.
Dogs that are allowed on furniture (even if put on the furniture) tend to feel that since they are elevated to your level or higher if on your lap, they mentally feel elevated as well in the pack order and thus are the boss. Keeping them on the floor can help lower them mentally back to a submissive position in the pack. So the first thing is to not allow him higher that the humans or even on the same levelAttach a leash and use it to remove him from the furniture. Give a correction in the form of a quick tug and firm "NO" when he attempts to get on and a treat when he starts not trying to get on the furniture. Thus you are providing negative reinforcement for the getting on the furniture and positive reinforcement for the desired behavior (not attempting to get on the furniture).
Additional training can also help. Both dogs should be leashed and if one dog even looks at the other dog, a correction should be done. Any sign of aggression including a prolonged look, hair raised on the shoulders, a growl or even a stiff legged walk, should be corrected. A correction is a quick tug of the leash and a firm low toned "NO". Once you have done this couple of times, you should notice the dogs ignoring each other. When that happens, you will want to reward them for the desired behavior. Again, use tasty treats like the hot dog slices. This teaches the dogs that you WILL not tolerate fighting in YOUR pack.
Once he is trained, you can start allowing him on the furniture if YOU ask him to get up but he has to get down when told to. Also pay attention to all the dogs' body language. It can give you cues as to who may be challenging the other and who is gettng ready to start issues which can allow you to stop fights before they start.
I hope this information is helpful to you. If you would like any additional information or have more questions please don’t hesitate to press the reply to expert or continue conversation button so I can address any issues you still have . If you do find this helpful, please take this opportunity to rate my answer so I am compensated for my time.