Strawberry,
I have reread the thread.
Firstly, I don't think anyone here (including me) is a doctor, and a doctor (and dietician) is what you really need. However, from experience and having done my own research, I do have some knowledge.
Weight, in and of itself, is not that important for health, until it really falls off the scale. Second centile is horribly skinny but not in any way life or development threatening (even at the 98th centile for height, which makes the effective "bmi" centile lower). What you say about his energy and development (when he is not sick) should be comforting to you. The lack of weight gain over 3 months is much more concerning and merits further investigation.
I hope I am not telling you something you already know but, when dealing with a skinny baby, you need to reverse the "health" rules for adults (or fat children) who need to worry about their calorie intake. In order of magnitude, starting at the top:
Fats: 1g=9 calories.
Simple sugars: 1g=4 calories
Protein: 1g=4 calories
Full fat milk: 1g=.66 calories
Complex carbs (rice, pasta, potatoes: 1g=1-1.5 calories.
The above are termed "macronutrients". There are also vitamins and minerals essential for metabolic processed within the body; these are termed "micronutrients".
I read a paper which suggested the absolute minimum a baby needs is 85 calories per kg per day. So, for your baby (who I am guessing weighs between 5 and 6 kg), he would need around 500 calories/day as an absolute minimum.
From what you say about your baby, he is getting around 140 calories from milk and around 100 calories from the rest (assuming your spoons are teaspoons). As you can see, this is half of what he needs. In addition, he will not be getting enough micronutrients.
Firstly, the micronutrient problem is soluble by supplementing with a decent vitamin supplement which you can buy over the counter. We used Abidec (and still do sometimes) but there is plenty of choice. Just chat to a pharmacist about what he/she would recommend. As for the macronutrients, you have to introduce more fats into the food. What kind of milk is he drinking. There is a formula (available on prescription) which is 100 calories/100ml. I seem to remember it is called "high energy". That is 50% higher than normal milk and is suitable for babies struggling to put on weight. Secondly, if he likes choc, cake, ice cream, give it to him! As you can see, you are getting twice as many calories per mouthful as just about anything else. Add cream and/or butter to pasta etc.
Sorry for the long post but we had very similar problems and I do feel for you.