1) Title: 'Impedance bump from coax shield grounding?'
Ideally, no. The RF is supposed to be ~inside~ the coaxial cable. The RF should 'have no idea' about what might be going on outside the shield.
In practice, RF is often coupled onto the outside of the coaxial cable and causes 'unbalanced currents' where the total shield current (inside + outside) is different than that on the center conductor.
Your 1:1 current balun is supposed to help in forcing the currents to be balanced. If the installation is not perfectly symmetrical, then the coax can have currents induced (onto the outside). A coil of coax can be used to choke-off such currents - you'll often see such coaxial baluns.
If there are currents on the outside of the coax (likely), then grounding the coax will cause those current paths to change. If the installation is ideal (unlikely), then it shouldn't cause any noticeable change.
Taking your post title even more literally (probably not your intent), the term 'impedance bump' strictly means that the characteristic impedance of the cable is changed at the point where the grounding block or lightning arrestor is inserted.
Such impedance bumps are usually not significant if they are physically small in terms of wavelength. Therefore, at HF frequencies (where the wavelengths are long), impedance bumps are usually considered to be a non-issue and are rarely mentioned. They are an issue on the higher bands.
All the above is just to respond to your title...
2) Towers responding differently to microwave and HF
An important concept to keep in mind is that a grounding point becomes a high impedance point only a quarter wavelength away. You can use a metal pipe as an insulator if it well grounded at one end and one quarter lamdba in length.
If your tower is significantly affecting your SWR, then find the new resonance point, calculate and adjust your dipole length to retune (if the SWR is an issue).
3) Lightning
Lightning and lightning protection is a topic that inspires much debate. There are many experts and even more that consider themselves 'experts'. I don't think that I'm an expert on the subject, but I've seen much obvious 'BS' from those that present themselves as experts. Be warned...
There is at least one point that is pretty much universally agreed: grounding is good, and the more the better.
But it is worth noting that commercial aircraft have excellent lightning protection, are typically hit about once per year (most often with no damage to their electronics), and are obviously not 'grounded' in the same manner as often recommended by the 'experts' for ground installations.
Obviously, there is more to lightning protection than just traditional earth grounding.
Putting your system into a metal box (airplane, commercial building) is an excellent first step. Then the concept of entrance protection can be made to work. The metal box can be flying 10,000 feet in the air, not grounded and the lightning protection system would still work.
Wooden building are totally different animals. Many experts do not acknoledge this important point when they recommend traditional entrance protection but fail to protect long signal wires inside the RF-transparent building.
Relative costs may also be an issue...