The original question mentions rightly that many systems can handle the mechanical requirements of mixing interactions between infantry and mechs. Older systems such as Palladium's Robotech and Rifts do this from a primarily abstract sense, as do newer systems like CthulhuTech and Revenants. In asking how to balance the interactions between meat and machine, the question seems to focus more on approaching these elements from a story or mind's eye sense than a tactical sense. It also seems to focus more on power armour than on mechs due to the use of the word suit. As a result, the connection to Battletech may not be immediately obvious. However, a game system which allows for all of the interactions described in the question, including power armour, is A Time of War; the 4th Edition of the venerable Mechwarrior franchise.
The Included Setting:
Battletech and its RPG formerly known as Mechwarrior envision a war torn future where the premier war machines are Mechs, 10-15m tall, piloted, generally anthropomorphic bi-pedal or quadrupedal, heavily armed and armoured weapons platforms capable of deployment virtually anywhere. The setting includes literally hundreds of these machines, plus vehicles, aircraft, naval ships, submarines, aerospace fighters, drop ships, small craft, jumpships, space-going warships, and all sorts of planetary and political environments in which humans find the need to battle.
It is a huge and sprawling setting, and it has been lovingly crafted and expanded for more than 25 years.
There are several eras represented in the setting from a golden technological age, through a long dark age of dwindling technology and enforced scavenging, back to a renaissance of discovery of old and new tech. There is a definite feel to the equipment of the setting. Mechs are not sentient nor are they controlled (until much, much later in the time line) by a direct neural linking or symbiosis between craft and pilot. (For that sort of thing check out CthulhuTech). Pilots control automated processes with sophisticated computer support, and provide the critical thinking and physical balance for the mech. Essentially, these are giant robots, controlled from within.
The RPG System:
The game system is quick to use, but can look daunting at first glance. It is based on rolling 2D6+modifiers for skills and actions. Many common interactions such as found in combat are actually quite elegant and are quick to pick up. The hardest part seems to be getting comfortable with the modifiers chart, and getting familiar with the equipment. There is a lot of equipment. In A Time of War you will find tech ranging from stone knives and bear skins on up to tailored biotoxins and anti-mech weaponry. The system for explaining the traits of all of this gear and all of these machines is consistent and once you start using it, makes a lot of sense. It is not something you can pick up and just play, but once you are playing, it works smoothly.
The A Time of War core book provides the RPG rules, as well as a tactical addendum for conducting play with the "zoom level" down to individual infantry vs mechs and other war machines. For abstracted play without maps, this and the technical readouts of the mechs themselves are all you need.
For play which includes broad and in-depth tactical miniatures use, the full rules for Battletech would be needed in addition to the RPG rules. The miniatures rules are found in Total Warfare and cover the mech zoom level as the default, with rules for infantry, aerospace, power armour, tanks and other armoured vehicles, as well as naval vessels.
Dropping the Setting:
Learning to run this game without its impressive and detailed setting would be quite simple. Adding AI to the mechs, or symbiosis between mech and pilot require no additional rules or tinkering, just imagination.
The concept of Operational Heat Build-up and how it affects and limits war machines in the rules may be something with which you would have to contend if this is not an aspect of mechs you wish to emulate in your games. It can simply be dropped, but if you are into designing your own machines via the design rules you will have to find a way to use the excess weight dropping this factor of the game would entail. Playing around with those ideas would be part of the fun. The mech design rules are found in the Techmanual.
To Infinity and Beyond?:
Additional zoom levels are available in the advanced rules, but these are purely optional and take play toward strategic levels of varying degrees, ultimately finishing on a galactic level of conquest. These zoom levels take the focus pretty firmly away from the man and machine interaction described in the question, so I will not detail them here.
Balancing Meat and Machine:
In Battletech the tactical war game of mech combat, and in its attached roleplaying game, the challenge of facing opponents is a primary concern, and the added challenges of using limited resources, facing superior or vastly superior forces are a major component of scenario design and dramatic tension. From a tactical sense, the onus is on the players to discover ways to tip the scales in their favour, or to use their resources appropriately. Can openers, not laser rifles, are for opening cans, but given a can-opener, how can one be used to foil the advance of a lance of mechs? For the GM, ensuring that the world can accommodate the plans of the players and allow them to be proactive is extremely important in getting players to bring their minds to bear on problems as much if not more than just the equipment of their characters.
The desired balance between in mech and out of mech tales can be supported through the use of the setting itself. The rules of warfare, rules and regulations of safe operations, and/or the threat of reaction from other mech-using forces can keep their deployment in check unless the circumstances call for it.
Resources can be a factor if you incorporate repair and maintenance costs, support crew requirements, transport, and so on. Wear and tear can be as big a threat to an operational force as weapons fire over time.
Scale, as always, will play the biggest role in guiding player decisions in interacting with the game world. If the players are not sensitive to appropriate scale of action (such as walking their mechs to a park to secretly meet an informant) then either this sensitivity will need to be increased, or the types of tale will need to be limited to focus on the sort of action the players seem to be desiring/fearing (combat).
It can sometimes be hard to expand the range of stories in an ostensibly mech-based game to out of mech activities, but working with the players to establish that the world is more than just battlefields and combat drops will go a long way to helping this to happen.