That's really not a problem
Keep in mind, for lack of full-size sedans, many police agencies are already going to full-size, truck-chassis SUVs, which are heavier than 1970s sedans. And handle worse since they are built on a truck chassis. (assuming both vehicles have received modern performance suspension tuning: the factory has bumped the truck suspension as far as it can, and the police force is fitting the best modern, performance aftermarket suspensions available.
Modernizing the engines is also not a problem. Modern engines are very high-performance, and can be retrofitted into older cars.
Keeping the engines up is really not a problem. A few modern engines will bolt right up to older chassis.
Let's take your Pontiac Lemans, e.g. the fourth generation (1973-77). That is a GM A-platform, meaning it had commonality with Chevelle, Malibu, etc. Rest assured we could do the same exercise on the B-platform or the C-platform.
Keeping up with the powertrains
The Lemans was already on a shared platform with cars from every other GM make - Chevy, Oldsmobile, Buick, Cadillac, even the GMC Sprint (an El Camino rebadging). And they put Chevy engines in some of those, so matching up the engine mounts is really not a problem.
Some of them took a Chevy small-block (302, 327, 350, 400). Those have always had their engine mounts the same shape and in the same places. That line was morphed into the LT-1 and LS-1 series, and I can tell you positively that the engine mounts did not move. I haven't bolted up an LS-3 or LS-7 but I have no reason to think they moved them. So an LS-7 engine will bolt right up to a Chevy Malibu or Impala police cruiser, as well as others of the A- B- C- platforms with an easy engine mount swap. All this stuff is readily available on the aftermarket, in better materials, no less.
Some of the truck engines could go over too, but since you said "CHP" I assume we have to pass smog. Needless to say the LS-7 package will pass 1979 smog numbers. The swap is allowed so long as it's complete and keeps the donor engine's emission system intact; and is equal or newer model year and class (car > light truck). It'll need a one-time inspection by CARB to affirm it was done correctly.
I don't know whether the same applies to the Mustang (whether a modern 4.6 will bolt into a '73 Mustang) but these are not hard problems for a customizer.
We'll have to keep the whole powertrain intact, actually.
The engine-transmission package is going to be pulled from the donor car fully dressed complete with engine computer, all sensors, even the fuel pump and parts of the dash. The engine will NOT be separated from the transmission!
The engine-transmission package will have no idea it's not in a 2015 Camaro.
The reason is that in a 2020 era car, the entire powertrain package works as a team. The PCM (Powertrain Control Module) tells the transmission when to shift, and the PCM manages the fuel injection through the "power transient" during that shift. That's done to hit the extremely agressive NZLEV smog numbers. (ironically Detroit's big V-8s were the first engines to meet the optional NZLEV standards, who would've expected that??) Even the throttle is "drive-by-wire" on some cars; again so the PCM has advance notice of the "power transient". Never over-injecting fuel is good for engine wear. That's why modern engines last longer.
One problem when comparing 1970s horsepower numbers to 1990s+ is that in the 1970s, the horsepower was based on the engine dynamometer figures - bare engine in a load cell, not even a tranny attached. However in the modern age, horsepower is full chassis "at the wheels" power turned in on a chassis dyno. That means a modern 300hp engine is actually quite a bit more powerful than a "classic" 300hp engine.
Speaking of that... for what it's worth, the 1973-77 LeMans was notable in that it shared its front suspension with the F-body Camaro/Firebird. There are plenty of aftermarket suspension upgrades for that!
The problem is that police cruisers need a lot of physical room. They can't be crammed into a Prius. As full-size sedans (Caprice, Crown Vic) have left the market, police have been forced into full sized SUVs like the Yukon. Well, the Yukon is built on the same chassis as the C1500 pickup truck, which itself has been "change-resistant" because pickup truck buyers do not like change. So it's a truck. It's gonna be doing like doing a police chase in a pickup truck. Granted it has all the performance tuning the modern age can give it, but still a truck. Not significantly better than what the modern age can give a 70s LeMans with a tuned Camaro front end.
However the LeMans has a much lower center of gravity and lower weight overall. (plus quite good forward-back balance with the lighter LS series engines in it).
Now, I know you can't outrun Mr. Motorola, but could such vehicles perform their intended law enforcement function in a modern world adequately? I'm talking Dodge Monacos and Polaras, the Pontiac LeMans, Plymouth Furies and Belvederes. Maybe even an ex-CHP Ford Mustang or two.
I feel confident that they could, yes.
A few things I would be doing in the story is upgrading the vehicle's police electronics--all would have modern two-way radios, Panasonic Toughbooks for the officers, and GPS location tracking.
Since you must bring over the PCM (Powertrain Control Module), you have the option to bring over the BCM (Body Control Module) also. That brings you as many of the modern creature comforts as you please to install, from OnStar to Siri to maps-in-the-dash. If it came in the donor car, you can bring it over.
So, how would a 1970's land yacht perform in traffic and patrol duties today? Could it work, or are Mopar squads just too old?
implying a Yukon isn't a land yacht
What else are they going to drive??? The best thing on the market is the last of the 2010s Crown Vics. Other than that, what most forces are going for is truck-derived SUVs because they need the space.