What IC should I use to make a binary clock

Depends on the flip-flop used in the shift register. Effectively, negate the enable line or assert the disable line :-), but you know that.

I'd ask, "Why not gate clocks, provided that you gate them 'properly'?"

A hazard is that if you asynchronously gate a clock you may (~= 50% chance for a symmetric clock) produce an additional clock transition at disable or re-enable time. ie DO NOT DO THIS

BUT if you gate synchronously so that eg when gate off the clock continues to its next high/low transition, then stays low while still gated off, then when gated on resumes on the next clock low/high edge.
As described that may allow a very narrow spike at either edge if gating signal just precedes a clock low/ high transition but a bit of common sense in implementation stops that.

Effectively if delay is acceptable between provided and utilised clock signal (and usually one clock cycle is indistinguishable from the next) then having a toggling FF in the clock line and disabling it will implement the above (possibly with the described spike hazard present.)

In this circuit from - FPGA Design tips

They say

• The following figure shows a synchronous alternative to the gated clock using a data path. The flip-flop is clocked at every clock cycle and the data path is controlled by an enable. When the enable is Low, the multiplexer feeds the output of the register back on itself. When the enable is High, new data is fed to the flip-flop and the register changes its state. This circuit guarantees a minimum clock pulse width and it does not add skew to the clock. The XC4000, Spartan-II, and Virtex families' flip-flops have a built-in clock-enable (CE).

Partially answering my "why not" question above - the gated clock will be delayed by the toggle propogation delay of the gating flip-flop. In some cases this does not matter, in others it's clearly ubbacceptable. The 3rd, worst case, is where it seems OK but sometimes just may not be, on an intermittent basis. So, use with care.

On this page Roadie Roger warns

• Actually gating clocks is relatively evil. Having a gated clock that transitions less often does save power. Every signal edge is a power waster in the face of capacitance. Gated clocks create other problems. Your clocks are no longer exactly lined up. Many chips have a global clock that goes everywhere. Breaking it into pieces isn't a good thing. You want to enable data rather than gate clocks. Call the Enable 'E'. This looks something like:

And Robert usefully counters

• Depends on your target technology. The above may hold for FPGA's, but for standard-cell based ASIC technologies, gated clocks are an elegant (in my opinion) and clean methodology for low-power design.

  In this case, you will most likely be using a clock tree generator anyway, and all of the clock tree generators I know easily handle clock gates. They will automatically align the clocks AFTER any possible gates, i.e. at the leaf nodes (clock inputs of registers). In this case, generating a low-skew clock tree is far easier for gated clocks than for derived clocks (i.e. divided clocks).

  Modern synthesis tools also can insert clock gates automatically by replacing register enables (as shown in the post above) by gated clocks. This can be handy, but it has the drawback of reducing your control over clock gates (names, hierarchy).

  Inserting clock gates in a VHDL design is easy. Just decide what gating style you want/need (and/or based, with/without latch, with test bypass, observability, ...).

  My choice is to then explicitly instantiate a clock gating element where needed. This has the disadvantage of requiring one to "route" different clock signals through the design, but it gives you full control over clock gating elements for use in later stages of the design (clock tree generation, physical optimization, ...).

  Information about "clean" clock gating can be found in books or also in synthesis tool manuals (where automatic clock gate insertion is described). Probably you'll also find resources on the web.

The sophistication of your clock gating depends on how much the downstream logic is going to respond to a bad clock pulse.

For example: If the downstream logic is always reset after the clocks switch then you can get away with almost anything for your clock gating logic.

However, if you are expecting the downstream logic to continue functioning properly when the clocks are switched (without a reset) then you gating logic must be designed so that you never have clock pulses that are too short and violate your timing constraints.

The danger with short clock pulses that violate your timing constraints is that it could put a state machine into an invalid state that it can't recover from. This is really easy to do, so don't take it for granted that it won't happen to you.

Designing good clock gating/switching logic that doesn't produce short clock pulses is difficult and usually requires intimate knowledge of your ASIC process. Rather than invent your own, I suggest that you talk with your ASIC house or whoever you got your ASIC libraries from. Odds are very high that they already have the circuit you need.

Update:

The OP indicated that the clock gates in the schematics are from the ASIC library. If those gates are designed to not glitch (I.E., they never produce a pulse that is too short), then the only trick is to allow enough time between disabling one clock and enabling the next. In this way you can cleanly switch between clocks and not have clock glitches.

Designing the logic to do this (keep enough time between disable/enable) is still difficult, but not super difficult. On a scale of 1-10, this would be a 6. Designing a glitchless clock gate, on the other hand, would be a 10 or even 12!