Hi Splash, Hmm... Kinda sounding like a blown head gasket. Start motor & see if it's leaking between cyl & head. Compression test will generally show low if blown. If just starting to blow it may show ok & only leak under power.
Valve lash is another word for valve clearance. Shop manual
calls it adjusting the valve rocker clearances. You can download shop manual
from any number of places. Classic British Spares is one.
Look at piston tops. What does top of pistons look like with flash light through plug hole? Lean & overheating will often show a whiteish piston top or tan, brown metal that looks like it may have been sand blasted with a very large sand. Most often Triumph piston tops look black & sooty & tend to look oily also often times.
I'd test compression before anything else to get a base line. Check cold & hot. Verify choke is off. Hold full throttle during kicking. Kick until gauge will read no higher. Count number of kicks & record. Reinstall plugs & start motor briefly on hot test, then check other side. Oil drains from rings very quickly during kicking & can lead to false low reading. Starting motor restores oil to rings. A good 1970 Tiger hot will read 170-175#. With carbon build up 180#. In my experience. Hot reading is the important reading. Triumph gave no specs.... Hot meaning after 5 miles or so.
I'd retorque head before valve adjustment.
If you've never done valve adjustment before it will take time & some practice to understand what you are doing. Feeler gauges will be very hard to use on your motor. I would use the angle of rotation of the screw. In any case when you are done you'll be able to wiggle rocker arm sideways slightly & feel some play. If it feels tight, it is tight. Tight will take much force to move sideways. Again the side play is only a little bit. Shop manual
explains how much a turn will give how much clearance. Do the math to see how much to turn for .004 & .002. Practice this with nut backed off slightly. Then just snug nut to until you can just turn screw. The reduces play in threads of screw. Tightening nut can change clearance so you need to practice & get a feel for it. Not rocket science, but does take practice.
Setting the motor position is most important. The key is turning rear wheel until the opposite valve is fully down. Use visual looking at valve & a finger on valve until you get the feel of fully down is. Then you're ready to check the opposite valve. Meaning when left exhaust valve is fully down (opened) you adjust the right valve. Then rotate motor until the right valve is down & adjust the left. Then move to intakes & do the same procedure.
Spec on 1970 Tiger is .004" on exhaust & .002" on intake.
VERY IMPORTANT!!! DO NOT turn the adjuster screw way out. If you do the push rod can come out of rocker arm & very hard to get back in. If you need to back off screw more than a turn or two to get clearance something is wrong. You need to put screw back. Tighten nut & verify the motor is in correct position to adjust that valve.
You don't need to split hairs but do the best you can.
Head retorque is strongly recommended. On rare occasion a loose head will blow smoke, but be good enough after torqueing.
Cold motor means after sitting overnight. Even slightly warm motor will skew adjustment greatly which can lead to burned valves.
Also what does top of pistons look like with flash light through plug hole? Lean & overheating will often show a whiteish piston top or tan, brown metal that looks like it may have been sand blasted with a very large sand. Most often Triumph piston tops look black & sooty & tend to look oily also often times. That is what I see on good running motors.
You have a lot on your plate here. Look over shop manual
& take some time to digest all this. No telling what you'll find.
You may need to purchase some tools. The valve adjuster nuts are close to 7/16" wrench but are actually BS/whitworth. I recommend you buy a set of sockets & combination wrenches. Triumph parts dealers or eBay
sells them. On a 1970 you'll need them if you get into working on this bike. You'll be surprised by where they are needed. You need torque wrench for head bolts. Get one that is appropriate for the head bolt torque specs. You need another for the larger fasteners like clutch nut. If you want problems, guess on torque.