Extra wrapping around a preformed cardboard tube is certainly possible and quite straightforward. I use a fairly strong wallpaper paste made up at 4.5g of paste to 100ml of water, weaker solutions often have problems with adhesion. For a 3/4" internal diameter tube with a wall thickness of 2.5mm, a 750mm strip of 90g/m2 Kraft paper will add about 1.5mm to the wall thickness and, of, course you can add as many layers as you need to achieve whatever wall thickness you like but, bear in mind that each subsequent layer will increase the diameter slightly less due to the increased circumference. I like to cut a width of paper that's about 40mm wider than the tube is long giving about 20mm extra at each end of the tube and allows plenty for trimming. I like to trim when the tube is dry enough, about 3 days for a 3mm extra wrapping using a plumbers plastic pipe cutter. A good allowance for trimming is necessary, if you go this way, as the tube needs to be supported on both sides of the blade and it is almost impossible to role the tubes without a degree of the wrappings running offline. I usually allow 7 days at room temperature to ensure they're dry and they should be stood on end whilst drying rather than laid on their side.
As for sodium silicate, it is used as a passive fireproofing material in building and construction so might work in pyro tubes. It's solubility is 22g per 100ml of water at 25 deg C so this is a guide to the maximum. However, solutions in water are commonly known as water glass and they're quite viscous, rather like Golden Syrup so you'll have to dilute it down to a more reasonable concentration to allow it to impregnate the cardboard. It's so viscous because the silicate anion is polymeric and, like all polymer solutions it's molecular weight produces significant viscosity. I have no experience of using it in the way you suggest and bow to the other posters who may have more experience, however, I do have one concern; its fire retardant properties come from the fact that it's an intumescent material. In a fire, the heat causes the silicate to release water of crystallisation as steam and the steam blows the coating up into a foam, the polymeric nature of the silicate anion stabilises the foam and prevents it breaking and the fact that it's a mineral means that it doesn't burn. Imagine something like a bath sponge made out of glass and you'll get the picture. All foams are great insulators so it prevents the heat of the fire reaching the substrate and thereby prevents further decomposition of the substrate with the release of flammable gasses so preventing combustion of the substrate. Intumescent materials can swell up by many times their original thickness, 50 - 100 times is not unusual so there is a slight risk of the silicate foam causing a blockage of the tube with unpleasant consequences. I'm not saying "don't" but be aware of the possibility and, first time out take suitable precautions.
Another way of fire proofing the inside of the tube that you might like to consider is using Parlon. Chlorinated hydrocarbons like Parlon were originally developed as fire retardants in the plastics industry and are still widely used as such. There's a certain irony, I think, about using a fire retardant as an important pyro ingredient but, that's the way of chemistry. Parlon has the advantage of being soluble in acetone so such a solution used to impregnate a tube would not suffer from the extended drying times that would be necessary for any water based treatment. I would make as concentrated solution as is possible from a practical point of view and apply several coats to the inside of the tube with a short drying time in between. Perhaps a longer drying time or drying somewhere warm would be advisable after the last coat.
Hope the above helps and I'd be interested to hear of any results.
