First, you are TOTALLY wrong regarding how much a 3.5db loss is. db is logarithmic. If you cut something by 3 db, you've cut it in half. If you've added 3db, you've doubled it. This is not "probably", it's fact.

The Decibel Measurement: A Ratio of Relative Strength - LiveAction
Second, yes, if you want to "make up" the db loss, you need to figure out the loss, then provide an amplifier to make up the loss. You are correct, my math was wrong. A four port splitter puts out ~25% of the power out of each output. Assuming your power starts at 100, that means each output power is at 25. So you need to make up (at least some of) that loss. If you simply double the input power (so +3 db), you now have "200" as your input level, but since each output is 25% of the input, your output level is 50. If you double the input power again (so now at +6 db), the input is now at 400, output is 25% of that, so 100.

Last, I'm really not suggesting the splitter has failed. But, it's no secret Dish's tuners aren't the best. If it is failing because of a lack of signal, the power loss from a splitter may be causing it. Bypassing the splitter and seeing if the situation improves is easy (a LOT easier than moving an antenna). Corrosion on connectors is another good check.

ETA: to change math. Grr. A 6db gain would quadruple (double + double) the signal strength.